Perfect Square & Square Root (Grade 7 Math)

Perfect Square & Square Root  OBJECTIVES define perfect square & square root determine the square roots of perfect squares  Perfect Square Explore Can you form a square with the given unit squares? a. 4 unit...

Perfect Square & Square Root  OBJECTIVES define perfect square & square root determine the square roots of perfect squares  Perfect Square Explore Can you form a square with the given unit squares? a. 4 unit c. 10 unit squares squares Yes No b. 9 unit d. 16 unit squares squares Yes Yes The number of unit squares that can form a square is called a perfect square.  Square Root Explore Can you find the length of the sides of a square if its area is given? a. b. c. d. Square s=1 s=2 s=3 s=4 s×s 1×1 2×2 3×3 4×4 Area 1 4 9 16 The square root of the area of the square (perfect square) is the length of the side of the square. Perfect Square & Square Root Perfect Square For Example: A perfect square is a number that is the product of a number multiplied by 1 itself. 4 In other words, it is a number that can be expressed as , read as “n squared”, 9 where n is an integer. 16 “n to the second power” or “n raised to the power of 2” 100  Square Root The square root of a number is the value that, when multiplied by itself, gives the original number. It is the inverse operation of squaring. The square root of a number m is written as , read as “the square root of m”, where m is called the radicand, is called the radical sign. For Example: The square root of 16 is 4 because 16 Radical Numbers The square root of 25 is 5 because 25  Activity A Complete the table below. Perfect Exponential Form Square Square (a number that when multiplied by itself, the answer is the number in column one) Root 9 3×3 3 36 6×6 6 49 7×7 7 81 9×9 9 121 11 × 11 11 625 25 × 25 25  Place each number in its appropriate column, Activity B then find the square root of the perfect square numbers: 0, 25, 40, 49, 121, 625, 8, 18/2, ¼, 27 Square Root of Perfect Not Perfect Square Perfect Square Number Square Numbers Number 0 0 40 25 5 8 49 7 27 121 11 625 25 18/2 =9 3 ¼ ½ How do you identify if a Method: -Trial and Error number is a perfect square? -Prime/factorization -Use Calculator -If it has an integer Square Root -Familiarization  Application Ms. Santos has a square garden with an area of 64 square meters. She wants to build a fence around the garden. How much fencing material will she need?  Up Next Perfect Cube and Cube Root Thank You for Listening Perfect Cube & Cube Root  OBJECTIVES define perfect cube & cube root determine the cube roots of perfect cube  Perfect Cube Explore Can you form a cube with the given unit cubes? a. 4 unit c. 16 unit cubes cubes No No b. 8 unit d. 27 unit cubes cubes Yes Yes The number of unit cubes that can form a cube is called a perfect cube.  Cube Root Explore Can you find the length of the side(edge) of a cube if its volume is given? a. b. c. d. Cube s=1 s=2 s=3 s=4 s×s×s 1×1×1 2×2×2 3×3×3 4×4×4 Volume 1 8 27 64 The cube root of the volume of a cube (perfect cube) is the length of each side(edge) of the cube Perfect Cube & Cube Root  Perfect Cube A perfect cube is a number that is For Example: obtained by multiplying the integer by 1 itself three times. In other words, it is a number that can 8 be expressed as , read as “n cubed”, 27 where n is an integer. 4 = 64 3 “n to the third power” or “n raised to the power of 3” 125  Cube Root The cube root of a number is the value that, when multiplied by itself three times, gives the original number. It is the inverse operation of cubing. The cube root of a number m is written as , read as “the cube root of m”, where m is called the radicand, is called the radical sign, and 3 is called the index. For Example: The cube root of 8 is 2 because 8 The cube root of 27 is 3 because 27  Activity A Complete the table below. Exponential Form Cube Perfect Cube (a number that when multiplied three times, the result is the given perfect cube) Root 1 1×1×1 1 8 2×2×2 2 125 5×5×5 5 216 6×6×6 6 1,000 10 × 10 × 10 10  Place each number in its appropriate column, Activity B then find the cube root of the perfect cube numbers: 27, 0, 9, 64, 81, 512, 729, 1/27, 343, 4/64 Cube Root of Perfect Not Perfect Cube Perfect Cube Number Cube Numbers Number 27 3 9 0 0 81 64 4 4/64 (1/16) 512 8 729 9 1/27 1/3 343 7  How do you identify if a number is a perfect cube? -If it has an integer Cube Root Method: -Trial and Error -Prime/factorization -Use Calculator -Familiarization  Application Mr. Dela Cruz has two cubic containers of different sizes. The larger of the two has sides that measure 10 cm, while the smaller one has sides that measure 8 cm. Will the two containers be enough for 500 cubic centimeters? Therefore, the two containers will be sufficient to hold 500 cubic centimeters.  Up Next Irrational Numbers Thank You for Listening Irrational Numbers  OBJECTIVES define irrational numbers determine the location of irrational numbers involving square roots and cube roots by plotting them on a number line. Rational Numbers Recap Fraction Percentage Decimal Whole number Proper Fraction Repeating Decimal Improper Fraction Terminating Decimal Mixed Number Non-Terminating & Non-Repeating Decimal  Examples of Irrational Numbers 4.1562547332… π (3.14159265...) Non-terminating & Non-repeating decimal e (2.71828...) (1.41421…)  Explore Place the following numbers in the appropriate columns: Rational Number Irrational Number 1/2 = 0.5 3 Irrational Numbers  Irrational Number Irrational numbers are numbers that cannot be expressed as a fraction of two integers (where the numerator and denominator are both whole numbers, and the denominator is not zero). In other words, they cannot be written in the form ​ , where p and q are integers.  Irrational Number If the radicand of a square root is not a perfect square, then it is considered an irrational number. Likewise, if the radicand of a cube root is not a perfect cube, then it is an irrational number. These numbers cannot be written as a fraction because the decimal does not end (or non-terminating) and does not repeat a pattern (or non-repeating).  Plotting Irrational Numbers  Plotting Irrational Numbers on the Number Line In plotting an irrational number involving square root or cube root on a number line, estimate first the square root or cube root of the given irrational number and to which two consecutive =1 & 4 integers it lies in between. identify two perfect squares nearest to the radicand 3. = 𝟏& 𝟒 For Example: locate and plot √3 on the number line 0 1 2 3 4 5  Plot the following irrational numbers Activity on the number line. 0 1 2 3 4 5 6 7 8 9 10 Thank You for Listening Grade 7 MATATAG Systems of Measure Part I Non-standard Units of Measure  OBJECTIVES identify the traditional or non-standard units of measure  Starter Discussion Questions: Have you ever measured an object before? What measuring tool and unit of measure have you used? What do you think is the role of measurement in everyday life? How do you think people measured things before rulers and measuring tapes existed?  Our Topic Systems of Measure Part I Part II Part III Non-standard English SI System Units of System of or Metric Measure Measurement System History of Measurement Non-standard Units of Measurement  Early Human Measurement body parts natural objects  Non-standard Units of Measurement A non-standard unit of measurement refers to any unit used for measuring that isn’t universally agreed upon or consistent across different users. “units of measure that aren’t fixed or consistent” EXAMPLES: Cubit It is the length from the elbow to the tip of the middle finger. In Ancient Egypt, a cubit is the In the Bible, the measures of length of a pharaoh’s arm. Noah’s Ark were given in cubits. EXAMPLES: Hand Span It is the distance from the tip of the thumb to the tip of the pinkie when the hand is fully extended. It was used for measuring cloth, as How many handspans do you the width was easily need to measure your desk? visualized by the human hand. EXAMPLES: Foot It was originally based on the length of an adult person’s foot. It was a common unit for measuring shorter lengths in everyday life and construction. EXAMPLES: Fathom A fathom equaled the span of a person’s outstretched arms. typically about six feet It was used to measure the depth of water and depth in nautical contexts. EXAMPLES: Footstep (Pace) The length of a single step taken by a person. often used informally to estimate distances soldiers used paces as a method of measuring distances on the battlefield or in formation drills.  Non-standard Units in the Philippines EXAMPLES: Kaban or Cavan The kaban is a traditional unit of dry measure used for rice. often equivalent to 50 kilograms it is not standardized and varies by region  Non-standard Units in the Philippines EXAMPLES: Dipper or Tabo It is often used for transferring water or other liquids and is not precisely standardized. In rural areas, people might use a dipper as an informal unit of measuring liquids.  Non-standard Units in the Philippines EXAMPLES: Tumpok This refers to a stack or pile of items, such as firewood, bananas, or coconuts. it is commonly used in daily life for selling and trading goods  Non-standard Units in the Philippines EXAMPLES: Hand Span Dangkal  Non-standard Units in the Philippines EXAMPLES: Foot Pace Fathom Cubit Talampakan Hakbang Dipa Bisig/Siko  Conclusion Discussion Questions: What are the advantages of using non-standard units of measurement? What are the disadvantages of using non-standard units of measurement? What other non-standard units of measurement do you know or practice?  Up Next Systems of Measure Part II English System of Measurement Thank You for Listening Grade 7 MATATAG Systems of Measure Part II English System of Measurement  OBJECTIVES identify the English System of Measurement  Our Topic Systems of Measure Part I Part II Part III Non-standard English SI System Units of System of or Metric Measure Measurement System  English System of Measurement  English System of Measurement It is also known as the Imperial System, a historical and culturally significant system of measurement that has been used primarily in the United States, Liberia, and Myanmar, and to a lesser extent, in the United Kingdom. History English System originated in England and was used throughout the British Empire before many countries switched to the metric system. The system was brought to the Philippines by the Americans during their colonization, but the Philippines now officially uses the metric system. Nowadays, many Filipinos continues to use this measurement system ENGLISH UNITS OF LENGTH Inches (in) 12 inches = 1 foot 3 feet = 1 yard Feet (ft) 5,280 feet = 1 mile Yards (yd) Miles (mi) ENGLISH UNITS OF LENGTH HISTORY Inches Feet Yards 1 yd 1 in 1 ft =12 in from nose to the tip of 3 grains of barley Roman soldier’s foot the outstretched arms It was said that in the 12th century, King Edward I of England standardized the inch, the foot and yard according to his own body measurement. ENGLISH UNITS OF WEIGHT Ounces (oz) 16 ounces = 1 pound 2,000 pounds = 1 ton Pounds (lb) Tons (t) ENGLISH UNITS OF VOLUME OR CAPACITY Teaspoon (tsp) Tablespoon (Tbsp) Fluid Ounces (fl oz) 3 teaspoon = 1 tablespoon 2 tablespoon = 1 fluid ounces Cups (c) 8 fluid ounces = 1 cup 16 tablespoon = 1 cup Gallons (gal) 16 cups = 1 gallon ENGLISH UNITS OF VOLUME OR CAPACITY usually 5 gallons These units are commonly used in United Kingdom and Ireland Pints (pt) 2 cups = 1 pint 2 pints = 1 quart Quarts (qt) 4 quarts = 1 gallon  Up Next Systems of Measure Part III SI System or Metric System of Measurement Thank You for Listening Grade 7 MATATAG Systems of Measure Part III SI/Metric System of Measurement  OBJECTIVES identify the SI or Metric System of Measurement  Our Topic Systems of Measure Part I Part II Part III Non-standard English SI System Units of System of or Metric Measure Measurement System  SI or Metric System of Measurement  SI (International System of Units) Also known as the Metric System, it is a globally accepted system of measurement that is decimal-based or based on powers of 10. The SI system is simple and standardized, allowing for easy conversion between units, making it the most widely used system in science, engineering, and everyday life worldwide. History During the French revolution, the metric system was formally adopted in France in 1795. The goal was to create a universal system that could be used by everyone, based on nature rather than arbitrary human standards. The Philippine government reaffirmed the use of the metric system through Republic Act in 2003. This act reinforced the exclusive use of the metric system in all commercial transactions, trade, and official matters in the country.  SI (International System of Units) The following are the metric The metric system prefixes and their meaning: uses prefixes to indicate what part of “kilo” - means 1,000 units the basic unit of “hecto” - means 100 units measure is being used. “deka” - means 10 units For example, in Base unit (1 unit) millimeter, the prefix “deci” - means 0.1 (1/10th of a unit) milli means “one thousandth” of a meter. “centi” - means 0.01 (1/100th of a unit) “milli” - means 0.001 (1/1,000th of a unit) SI/METRIC UNITS OF LENGTH Meter (m) SI/METRIC UNITS OF LENGTH Meter (m) SI/METRIC UNITS OF MASS/WEIGHT Gram (g) SI/METRIC UNITS OF VOLUME OR CAPACITY Liter (L)  SI (International System of Units) Other metric prefixes and their meaning:  Conclusion Discussion Questions: What are the advantages of using standard units of measurement? What are the disadvantages of using standard units of measurement? What other standard units of measurement do you know or practice? Thank You for Listening Grade 7 MATATAG Conversion of Units Part I Metric System Conversion kg hg dag g dg cg mg  OBJECTIVES convert units of measurement within the metric system solve word problems involving the conversion of units  Recap SI (International System of Units) Also known as the Metric System, it is a globally accepted system of measurement that is decimal-based or based on powers of 10. “kilo” “hecto” “deka” Base unit “deci” “centi” “milli” 1000 100 10 (1 unit) 0.1 0.01 0.001 Metric System to Metric System Conversion kg hg dag g dg cg mg  Units of Length EXAMPLE #1: Convert 7.25 meters to centimeters. Method #1: using a conversion factor/unit fraction Conversion Factor: Unit Fraction: 1m 100 cm 1 m = 100 cm or 100 cm 1m Solution: 100 cm 7.25 m × = 725 cm 1m  Units of Length EXAMPLE #1: Convert 7.25 meters to centimeters. Method #2: by moving the decimal point km hm dam m dm cm mm 2 units to the right Solution: 7.25 m = 725 cm 2 places to the right  Units of Mass/Weight EXAMPLE #2: Convert 4800 grams to kilograms. Method #1: using a conversion factor/unit fraction Conversion Factor: Unit Fraction: 1 kg = 1000 g 1 kg 1000 g Solution: 1 kg 4800 g × = 4.8 kg 1000 g  Units of Mass/Weight EXAMPLE #2: Convert 4800 grams to kilograms. Method #2: by moving the decimal point kg hg dag g dg cg mg 3 units to the left Solution: 4800 g = 4.8 kg 3 places to the left  Units of Volume EXAMPLE #3: Convert 2.25 liters to milliliters. Method #1: using a conversion factor/unit fraction Conversion Factor: Unit Fraction: 1 L = 1000 mL 1000 mL 1L Solution: 1000 mL 2.25 L × = 2250 mL 1L  Units of Volume EXAMPLE #3: Convert 2.25 liters to milliliters. Method #2: by moving the decimal point kL hL daL L dL cL mL 3 units to the right Solution: 2.250 L = 2250 L 3 places to the right Solving Word Problems Involving Metric Unit Conversion  Solving Word Problems Involving Metric Unit Conversion EXAMPLE #1: Your cousin is planning a hike, and the trail map lists distances in kilometers (km). However, they want to know the distance in meters (m) to better track their progress. Help them convert 7.1 kilometers to meters. Method #1: Method #2: Conversion Factor: 1 km = 1000 m km hm dam m dm cm mm Solution: Solution: 1000 m 7.1 km × = 7100 m 7.100 km = 7100 m 1 km  Solving Word Problems Involving Metric Unit Conversion EXAMPLE #2: You are hosting a family reunion and preparing a big pot of traditional Filipino soup that requires 4 liters (L) of broth. However, your measuring tools are in milliliters (mL). Convert 4 liters of broth to milliliters. Method #1: Method #2: Conversion Factor: 1 L = 1000 mL kL hL daL L dL cL mL Solution: Solution: 1000 mL 4L × = 4000 mL 4 000 L = 4000 mL 1L  Solving Word Problems Involving Metric Unit Conversion EXAMPLE #3: You are helping your grandmother manage her medications. She needs to take 1.2 grams (g) of a certain medicine every day, but the tablets she has are only available in milligrams (mg). How many milligrams of the medicine should she take each day? Method #1: Method #2: Conversion Factor: 1 g = 1000 mg kg hg dag g dg cg mg Solution: Solution: 1000 mg 1.2 g × = 1200 mg 1.2 00 g = 1200 mg 1g  Solve the following metric Activity A conversion problems 1. A string is 56 centimeters long. Convert this length to millimeters. 2.A bag of flour weighs 1.75 kilograms. Convert this weight to milligrams. 3. A water bottle contains 625 milliliters of water. Express this volume in liters.  Solve the following metric Activity B conversion problems 1. Your friend is training for a marathon and tracks their running distance in kilometers (km). To calculate how far they’ve run in meters (m), convert 6.5 kilometers to meters. 2.You are making fruit juice for a community gathering, and the recipe calls for 5 liters (L) of water. Your container only has measurements in milliliters (mL). Convert 5 liters to milliliters. 3. You are packing groceries for a delivery service, and the vegetables are weighed in kilograms (kg). You need to print out the weight in grams (g) for the receipt. Convert 4.5 kilograms to grams. Activity A SOLUTIONS: A string is 56 centimeters long. Convert this length to millimeters. km hm dam m dm cm mm Activity A SOLUTIONS: A bag of flour weighs 1.75 kilograms. Convert this weight to milligrams. kg hg dag g dg cg mg Activity A SOLUTIONS: A water bottle contains 625 milliliters of water. Express this volume in liters. kL hL daL L dL cL mL Activity B SOLUTIONS: Your friend is training for a marathon and tracks their running distance in kilometers (km). To calculate how far they’ve run in meters (m), convert 6.5 kilometers to meters. km hm dam m dm cm mm  Activity B SOLUTIONS: You are making fruit juice for a community gathering, and the recipe calls for 5 liters (L) of water. Your container only has measurements in milliliters (mL). Convert 5 liters to milliliters. kL hL daL L dL cL mL Activity B SOLUTIONS: You are packing groceries for a delivery service, and the vegetables are weighed in kilograms (kg). You need to print out the weight in grams (g) for the receipt. Convert 4.5 kilograms to grams. kg hg dag g dg cg mg  Up Next Conversion of Units Part II English System Conversion Thank You for Listening Grade 7 MATATAG Conversion of Units Part I Metric System Conversion kg hg dag g dg cg mg  OBJECTIVES convert units of measurement within the metric system solve word problems involving the conversion of units  Recap SI (International System of Units) Also known as the Metric System, it is a globally accepted system of measurement that is decimal-based or based on powers of 10. “kilo” “hecto” “deka” Base unit “deci” “centi” “milli” 1000 100 10 (1 unit) 0.1 0.01 0.001 Metric System to Metric System Conversion kg hg dag g dg cg mg  Units of Length EXAMPLE #1: Convert 7.25 meters to centimeters. Method #1: using a conversion factor/unit fraction Conversion Factor: Unit Fraction: 1m 100 cm 1 m = 100 cm or 100 cm 1m Solution: 100 cm 7.25 m × = 725 cm 1m  Units of Length EXAMPLE #1: Convert 7.25 meters to centimeters. Method #2: by moving the decimal point km hm dam m dm cm mm 2 units to the right Solution: 7.25 m = 725 cm 2 places to the right  Units of Mass/Weight EXAMPLE #2: Convert 4800 grams to kilograms. Method #1: using a conversion factor/unit fraction Conversion Factor: Unit Fraction: 1 kg = 1000 g 1 kg 1000 g Solution: 1 kg 4800 g × = 4.8 kg 1000 g  Units of Mass/Weight EXAMPLE #2: Convert 4800 grams to kilograms. Method #2: by moving the decimal point kg hg dag g dg cg mg 3 units to the left Solution: 4800 g = 4.8 kg 3 places to the left  Units of Volume EXAMPLE #3: Convert 2.25 liters to milliliters. Method #1: using a conversion factor/unit fraction Conversion Factor: Unit Fraction: 1 L = 1000 mL 1000 mL 1L Solution: 1000 mL 2.25 L × = 2250 mL 1L  Units of Volume EXAMPLE #3: Convert 2.25 liters to milliliters. Method #2: by moving the decimal point kL hL daL L dL cL mL 3 units to the right Solution: 2.250 L = 2250 L 3 places to the right Solving Word Problems Involving Metric Unit Conversion  Solving Word Problems Involving Metric Unit Conversion EXAMPLE #1: Your cousin is planning a hike, and the trail map lists distances in kilometers (km). However, they want to know the distance in meters (m) to better track their progress. Help them convert 7.1 kilometers to meters. Method #1: Method #2: Conversion Factor: 1 km = 1000 m km hm dam m dm cm mm Solution: Solution: 1000 m 7.1 km × = 7100 m 7.100 km = 7100 m 1 km  Solving Word Problems Involving Metric Unit Conversion EXAMPLE #2: You are hosting a family reunion and preparing a big pot of traditional Filipino soup that requires 4 liters (L) of broth. However, your measuring tools are in milliliters (mL). Convert 4 liters of broth to milliliters. Method #1: Method #2: Conversion Factor: 1 L = 1000 mL kL hL daL L dL cL mL Solution: Solution: 1000 mL 4L × = 4000 mL 4 000 L = 4000 mL 1L  Solving Word Problems Involving Metric Unit Conversion EXAMPLE #3: You are helping your grandmother manage her medications. She needs to take 1.2 grams (g) of a certain medicine every day, but the tablets she has are only available in milligrams (mg). How many milligrams of the medicine should she take each day? Method #1: Method #2: Conversion Factor: 1 g = 1000 mg kg hg dag g dg cg mg Solution: Solution: 1000 mg 1.2 g × = 1200 mg 1.2 00 g = 1200 mg 1g  Solve the following metric Activity A conversion problems 1. A string is 56 centimeters long. Convert this length to millimeters. 2.A bag of flour weighs 1.75 kilograms. Convert this weight to milligrams. 3. A water bottle contains 625 milliliters of water. Express this volume in liters.  Solve the following metric Activity B conversion problems 1. Your friend is training for a marathon and tracks their running distance in kilometers (km). To calculate how far they’ve run in meters (m), convert 6.5 kilometers to meters. 2.You are making fruit juice for a community gathering, and the recipe calls for 5 liters (L) of water. Your container only has measurements in milliliters (mL). Convert 5 liters to milliliters. 3. You are packing groceries for a delivery service, and the vegetables are weighed in kilograms (kg). You need to print out the weight in grams (g) for the receipt. Convert 4.5 kilograms to grams. Activity A SOLUTIONS: A string is 56 centimeters long. Convert this length to millimeters. km hm dam m dm cm mm Activity A SOLUTIONS: A bag of flour weighs 1.75 kilograms. Convert this weight to milligrams. kg hg dag g dg cg mg Activity A SOLUTIONS: A water bottle contains 625 milliliters of water. Express this volume in liters. kL hL daL L dL cL mL Activity B SOLUTIONS: Your friend is training for a marathon and tracks their running distance in kilometers (km). To calculate how far they’ve run in meters (m), convert 6.5 kilometers to meters. km hm dam m dm cm mm  Activity B SOLUTIONS: You are making fruit juice for a community gathering, and the recipe calls for 5 liters (L) of water. Your container only has measurements in milliliters (mL). Convert 5 liters to milliliters. kL hL daL L dL cL mL Activity B SOLUTIONS: You are packing groceries for a delivery service, and the vegetables are weighed in kilograms (kg). You need to print out the weight in grams (g) for the receipt. Convert 4.5 kilograms to grams. kg hg dag g dg cg mg  Up Next Conversion of Units Part II English System Conversion Thank You for Listening Grade 7 MATATAG Volume of Cylinder  OBJECTIVES explain inductively the volume of a cylinder using the area of a circle, leading to the identification of the formula find the volume of a cylinder  Lesson Sequence Recap: Area and Volume Area of a Circle Lesson: Derivation of Volume of a Cylinder Solving Volume of a Cylinder  Area Volume Recap 2D 3D “flat shape” “solid objects” square units cubic units A=l×w V=l×w×h  Recap Area of a Circle  Recap How is the area of circle formula derived? https://www.geogebra.org/m/enx6nxqx Deriving Volume of a Cylinder  Cylinder Volume A cylinder is a three-dimensional geometric shape with two parallel and congruent circular bases connected by a curved surface. How is the volume of a cylinder formula derived? https://www.geogebra.org/m/vaunxfeq Volume of a Cylinder Solving Volume of a Cylinder  Solving Volume of a Cylinder Example 1. Find the volume of a cylinder with radius measuring 2 meters and height 4 meters. (Use 𝜋=3.14) Solution: 2 𝑉𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟=𝜋𝑟 ℎ ℎ=4𝑚 2 𝑉= (3.14)(2 )(4) 𝑉 = (3.14)(4)(4) 𝑉 = (12.56)(4) 𝑟 =2𝑚 3 The volume of the 𝑉= 50.24 𝑚 cylinder is 50.24 𝑚3  Solving Volume of a Cylinder 𝑑 = 5 𝑐𝑚 Example 2. The diameter of a can of sardines is 5 𝑐𝑚, and the height is 8.5 𝑐𝑚. What is the volume of the tin can? Solution: 2 𝑉𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟 = 𝜋𝑟 ℎ 𝑉 = (3.14)(2.5 ) 2 (8.5) 𝑑 𝑉 = (3.14)(6.25)(8.5) 𝑟= 2 5 𝑉 = (19.625)(8.5) 𝑟= 2 𝑉 = 166.8125 c𝑚 3 ℎ = 8.5 𝑐𝑚 𝑟 = 2.5 The volume of the tin ≈ 166.81 𝑐𝑚 3 can is 166.81 𝑐𝑚3  Solving Volume of a Cylinder Example 3. What is the height of a cylinder whose radius is 5 𝑑𝑚 and has a volume of 471 𝑑𝑚 ? 3 Solution: r = 5 𝑑𝑚 𝑉𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟 = 𝜋𝑟 ℎ2 471 = (3.14) 5 2 ℎ Given: 471 = (3.14) 25 ℎ 𝑉 = 471 𝑑𝑚 3 471 = (78.5)ℎ 𝑟 = 5 𝑑𝑚 471 = ℎ ℎ=? 78.5 ℎ = 6 𝑑𝑚 The height of the cylinder is 6 𝑑𝑚  Solving Volume of a Cylinder Example 4. What is the radius of the cylinder whose height is 4 𝑓𝑡 and volume of 50.24 𝑓𝑡 ? 3 Solution: ℎ = 4 𝑓𝑡 2 𝑉𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟 = 𝜋𝑟 ℎ 50.24 = (3.14) 𝑟 2 (4) Given: 50.24 = (12.56)𝑟 2 3 50.24 𝑉 = 50.24 𝑓𝑡 = 𝑟 2 2=𝑟 12.56 ℎ = 4 𝑓𝑡 2 The radius of the 4 = 𝑟 𝑟 = 2 𝑓𝑡 𝑟=? cylinder is 2 𝑓𝑡 4=𝑟 2  Activity Find the volume of the solid figure with the following measurements: 1. A water tank has a radius of 3 meters and a height of 6 meters. Find the volume of the water tank. (Use 𝜋 = 3.14) 2.The diameter of a soup can is 6 cm, and the height is 10 cm. What is the volume of the can? 3. A flower pot in the shape of a cylinder has a radius of 4 decimeters and a volume of 502.4 cubic decimeters. What is the height of the flower pot? 4.A rice container in the shape of a cylinder is used by a sari-sari store. The height of the container is 1 meter, and the volume is 157 cubic meters. What is the radius of the container? Activity SOLUTIONS: 1. A water tank has a radius of 3 meters and a height of 6 meters. Find the volume of the water tank. (Use 𝜋=3.14) The volume of the water tank is 169.56 cubic meters. Activity SOLUTIONS: 2. The diameter of a soup can is 6, and the height is 10. What is the volume of the can? The volume of the soup can is 282.6 cubic centimeters. Activity SOLUTIONS: 3. A flower pot in the shape of a cylinder has a radius of 4 decimeters and a volume of 502.4 cubic decimeters. What is the height of the flower pot? The height of the flower pot is 10 decimeters.  Activity SOLUTIONS: 4. A rice container in the shape of a cylinder is used by a sari-sari store. The height of the container is 1 meter, and the volume is 157 cubic meters. What is the radius of the container? The radius of the rice container is approximately 7.07 meters.  Up Next Volume of Cylinder Solving Problems Thank You for Listening Grade 7 MATATAG Solving Problems: Volume of Cylinder  OBJECTIVES solve problems involving the volumes of cylinders  Recap Area of a Circle Volume of a Cylinder Solving Volume of a Cylinder  Solving Volume of a Cylinder Example 1. A cylindrical drum is used to store rainwater for a small household. The drum has a radius of 1.2 meters and a height of 3 𝑚𝑒𝑡𝑒𝑟𝑠. What is the total volume of water the drum can hold? (Use 𝜋=3.14) Given: Solution: 2 𝑟 = 1.2𝑚 𝑉𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟=𝜋𝑟 ℎ 2 ℎ=3𝑚 𝑉 = (3.14)(1.2 )(3) 𝑉 = (3.14)(1.44)(3) 𝑉=? 𝑉 = (3.14)(4.32) 𝑉 = 13.5648 The volume of 3 the cylinder is ≈ 13.56 𝑚 13.56 𝑚3  Solving Volume of a Cylinder Example 2. A farmer uses a cylindrical silo to store grain. The silo has a diameter of 10 𝑓𝑒𝑒𝑡 and a height of 15 𝑓𝑒𝑒𝑡. What is the total volume of the silo in cubic feet? Given: Solution: 𝑉𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟=𝜋𝑟 2 ℎ ℎ = 15 𝑓𝑡 𝑑 2 𝑟= 𝑉 = (3.14)(5 )(15) 2 𝑑 =10 𝑓𝑡 10 𝑉 = (3.14)(25)(15) 𝑟=? 𝑟= 𝑉 = (3.14)(375) 2 𝑉=? 𝑟=5 𝑉= 1,177.5 𝑓𝑡 3 The volume of the 3 cylinder is 1,177.5 𝑓𝑡  Solving Volume of a Cylinder Example 3. A large cylindrical water tank can hold 904.32 𝑚 of water. The height of the tank is 6 𝑚. What is 3 the radius of the tank? Given: Solution: 3 𝑉𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟 = 𝜋𝑟 2 ℎ 𝑉 =904.32 𝑚 ℎ=6𝑚 904.32 = (3.14) 𝑟 2 (6) 2 𝑟=? 904.32 = (18.84)𝑟 904.32 2 =𝑟 18.84 6.92820… = 𝑟 2 𝑟 = 6.92820... 48 = 𝑟 The radius of the 48 = 𝑟 2 ≈ 6.93 𝑚 cylinder is 6.93 𝑚  Solving Volume of a Cylinder Example 4. A cylindrical propane tank has a radius of 2 𝑓𝑡 and a volume of 50.24 𝑓𝑡. What is the height of the 3 tank? Given: Solution: 3 2 𝑉𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟 = 𝜋𝑟 ℎ 𝑉 = 50.24 𝑓𝑡 𝑟 = 2 𝑓𝑡 50.24 = (3.14) 2 ℎ 2 ℎ=? 50.24 = (3.14) 4 ℎ 50.24 = (12.56)ℎ 50.24 =ℎ 12.56 ℎ = 4 𝑓𝑡 The height of the cylinder is 4 𝑓𝑡.  Solving Volume of a Cylinder Example 5. A cylindrical water container needs to be filled with water for a barangay event. The container has a height of 2.5 𝑚 and a radius of 1 𝑚. However, due to a leak, the container can only hold 80% of its total volume. How much water can the container hold in cubic meters? Given: Solution: 𝑟=1𝑚 𝑉𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟=𝜋𝑟 ℎ2 𝑉water = 7.85 × 80% 2 ℎ = 2.5 𝑚 𝑉 = (3.14)(1 )(2.5) 𝑉water = (7.85)(0.80) 𝑉 = (3.14)(1)(2.5) 𝑉water = 6.28 𝑚 3 𝑉=? 𝑉 = 7.85 𝑚 3 The container can hold 6.28 m3 of water due to the leak  Activity Find the volume of the solid figure with the following measurements: 1. A cylindrical drum is used to collect rainwater for irrigation in a community garden. The drum has a radius of 1.5 𝑚𝑒𝑡𝑒𝑟𝑠 and a height of 2.5 𝑚𝑒𝑡𝑒𝑟𝑠. What is the total volume of water the drum can hold in cubic meters? (𝑈𝑠𝑒 𝜋 = 3.14) 2.A farmer uses a cylindrical silo to store hay for livestock. The silo has a diameter of 12 𝑓𝑒𝑒𝑡 and a height of 10 𝑓𝑒𝑒𝑡. What is the total volume of the silo in 𝑐𝑢𝑏𝑖𝑐 𝑓𝑒𝑒𝑡? 3. A large cylindrical water tank is used to store water for a barangay. The tank can hold 1,256 𝑐𝑢𝑏𝑖𝑐 meters of water. The height of the tank is 8 𝑚𝑒𝑡𝑒𝑟𝑠. What is the radius of the tank?  Activity Find the volume of the solid figure with the following measurements: 4. A cylindrical propane tank has a radius of 3 𝑓𝑡 and a volume of 84.78 𝑓𝑡 3. What is the height of the tank? 5. A cylindrical water container is used to store water for a school event. The container has a height of 5 𝑚 and a radius of 2 𝑚. However, due to a leak, the container can only hold 75% of its total volume. How much water can the container hold in cubic meters?  Activity SOLUTIONS: 1. A cylindrical drum is used to collect rainwater for irrigation in a community garden. The drum has a radius of 1.5 𝑚𝑒𝑡𝑒𝑟𝑠 and a height of 2.5 𝑚𝑒𝑡𝑒𝑟𝑠. What is the total volume of water the drum can hold in cubic meters? (𝑈𝑠𝑒 𝜋 = 3.14) Solution: 2 𝑉𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟=𝜋𝑟 ℎ 2 𝑉 = 17.6625 The drum can hold 𝑉 = (3.14)(1.5 )(2.5) approximately 3 𝑉 = (3.14)(2.25)(2.5) ≈ 17.66 𝑚 17.66 𝑚3 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟. 𝑉 = (3.14)(5.625) Activity SOLUTIONS: 2. A farmer uses a cylindrical silo to store hay for livestock. The silo has a diameter of 12 𝑓𝑒𝑒𝑡 and a height of 10 𝑓𝑒𝑒𝑡. What is the total volume of the silo in 𝑐𝑢𝑏𝑖𝑐 𝑓𝑒𝑒𝑡? Solution: 𝑉𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟=𝜋𝑟 ℎ 2 𝑑 2 𝑟= 𝑉 = (3.14)(6 )(10) 2 12 𝑉 = (3.14)(36)(10) 𝑟= 2 𝑉 = (3.14)(360) 𝑟=6 3 The silo can hold approximately 𝑉 = 1,130.4 𝑓𝑡 3 1,130.4 𝑓𝑡 of hay.  Activity SOLUTIONS: 3. A large cylindrical water tank is used to store water for a barangay. The tank can hold 1,256 𝑐𝑢𝑏𝑖𝑐 meters of water. The height of the tank is 8 𝑚𝑒𝑡𝑒𝑟𝑠. What is the radius of the tank? 𝑉𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟 = 2 𝜋𝑟 ℎ 7.071067… = 𝑟 1,256 = (3.14) 𝑟 (8) 2 𝑟 = 7.071067… 2 1,256 = (25.12)𝑟 ≈7.07 𝑓𝑡 1,256 2 = 𝑟 The radius of the 25.12 cylinder is 7.07 𝑓𝑡 2 50 = 𝑟 2 50 = 𝑟  Activity SOLUTIONS: 4. A cylindrical propane tank has a radius of 3 𝑓𝑡 and a volume of 84.78 𝑓𝑡 3. What is the height of the tank? Solution: 𝑉𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟 = 2 𝜋𝑟 ℎ 84.78 = (3.14) 3 ℎ 2 84.78 = (3.14) 9 ℎ 84.78 = (28.26)ℎ 84.78 =ℎ ℎ = 3 𝑓𝑡 The height of the cylinder is 3 𝑓𝑡. 28.26  Activity SOLUTIONS: 5. A cylindrical water container is used to store water for a school event. The container has a height of 5 𝑚 and a radius of 2 𝑚. However, due to a leak, the container can only hold 75% of its total volume. How much water can the container hold in cubic meters? Solution: 𝑉𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟=𝜋𝑟 ℎ2 𝑉water = 62.8 × 75% 2 𝑉 = (3.14)(2 )(5) 𝑉water = (62.8)(0.75) 𝑉 = (3.14)(4)(5) 𝑉water = 47.1 𝑚 3 3 𝑉 = 62.8 𝑚 The container can hold 47.1 𝑚3 of water due to the leak Thank You for Listening Grade 7 MATATAG Volume of Rectangular Pyramid  OBJECTIVES explain inductively the volume of a rectangular pyramid, leading to the identification of the formula find the volume of a rectangular pyramid  Lesson Sequence Recap: Area of a Rectangle Volume of a Rectangular Prism Lesson: Derivation of Volume of a Rectangular Pyramid Solving Volume of a Rectangular Pyramid  Recap Area of a Rectangle w=4 l=7 A = (7)(4) 28 square units  Recap Volume of a Rectangular Prism https://www.geogebra.org/m/rp9qga8c Deriving Volume of a Pyramid  Pyramid A pyramid is a three- dimensional solid with a polygonal base and triangular faces that converge to a single point called the apex.  Rectangular Pyramid A rectangular pyramid is a type of pyramid where the base is a rectangle. It has four triangular faces that meet at the apex, and each triangular face is connected to one side of the rectangular base. How is the volume of a rectangular pyramid derived? https://www.geogebra.org/m/gkuka5cj How is the volume of a rectangular pyramid derived? Volume of a Rectangular Pyramid  Solving Volume of a Rectangular Pyramid  Solving Volume of a Rectangular Pyramid Example 1. A rectangular pyramid whose base area is 120 𝑓𝑡 and height of 9 𝑓𝑡. Find the 2 volume of the pyramid. Solution: 1 𝑉pyramid = 𝐵ℎ 3 1 𝑉 = (120)(9) ℎ = 9 𝑓𝑡 3 1 𝑉= 3 (1080) 3 The volume of the 𝑉= 360 𝑓𝑡 2 rectangular pyramid 𝐵 = 120 𝑓𝑡 is 360 𝑓𝑡 3  Solving Volume of a Rectangular Pyramid Example 2. Find the volume of a rectangular pyramid if its base length is 10 𝑚𝑒𝑡𝑒𝑟𝑠, its base width is 6 𝑚𝑒𝑡𝑒𝑟𝑠, and its height is 14 𝑚𝑒𝑡𝑒𝑟𝑠. Solution: 1 𝑉pyramid = 𝑙𝑤ℎ 3 1 𝑉 = (10)(6)(14) = 14 𝑚 3 1 𝑉= 3 (840) 3 The volume of the =6𝑚 𝑉= 280 𝑚 rectangular pyramid = 10 𝑚 is 280 𝑚3  Solving Volume of a Rectangular Pyramid Example 3. A rectangular pyramid has a base length of 10 𝑓𝑒𝑒𝑡 and a width of 7 𝑓𝑒𝑒𝑡. The height is approximately 9.5 𝑓𝑒𝑒𝑡. Estimate the volume of the pyramid. Solution: 1 𝑉pyramid = 𝑙𝑤ℎ 3 ℎ = 9.5 𝑓𝑡 1 𝑉 = (10)(7)(9.5) 3 1 𝑉= 3 (665) 3 𝑤 = 7 𝑓𝑡 𝑉= 221.67 𝑓𝑡 The volume of the rectangular pyramid 𝑙 = 10 𝑓𝑡 is 221.67 𝑓𝑡 3  Solving Volume of a Rectangular Pyramid Example 4 The volume of a rectangular pyramid is 300 𝑚 , 3 and the base area is 50 𝑚. Find the height of the pyramid. 2 50 50 Given: 300 = ℎ 300 = 3 ℎ 3 3 50 V = 300 𝑚 300 = 16.67ℎ 300(3) = (3)ℎ 2 300 3 B = 50 𝑚 =ℎ 16.67 900 = 50ℎ Solution: 17.9964 … = ℎ 900 1 = ℎ 𝑉pyramid = 𝐵ℎ ≈ 18 50 3 1 ℎ = 18 𝑚 ℎ = 18 𝑚 300 = (50)ℎ 3 The height of the rectangular pyramid is 18 m  Activity Find the volume of the solid figure with the following measurements: 1. A rectangular pyramid has a base area of 150 𝑓𝑡 2 and a height of 12 𝑓𝑡. Find the volume of the pyramid. 2.Find the volume of a rectangular pyramid if its base length is 8 𝑚𝑒𝑡𝑒𝑟𝑠, its base width is 5 𝑚𝑒𝑡𝑒𝑟𝑠, and its height is 16 𝑚𝑒𝑡𝑒𝑟𝑠. 3. A rectangular pyramid has a base length of 9 feet and a width of 6 feet. The height is approximately 10.5 feet. Estimate the volume of the pyramid. 4.The volume of a rectangular pyramid is 450 𝑚 , and the base area 3 is 75 𝑚2. Find the height of the pyramid.  Activity SOLUTIONS: 1. A rectangular pyramid has a base area of 150 𝑓𝑡 2 and a height of 12 𝑓𝑡. Find the volume of the pyramid. The volume of the pyramid is 600 cubic feet.  Activity SOLUTIONS: 2. Find the volume of a rectangular pyramid if its base length is 8 𝑚𝑒𝑡𝑒𝑟𝑠, its base width is 5 𝑚𝑒𝑡𝑒𝑟𝑠, and its height is 16 𝑚𝑒𝑡𝑒𝑟𝑠. Rounding to the second decimal place, the volume of the pyramid is 213.33 cubic meters.  Activity SOLUTIONS: 3. A rectangular pyramid has a base length of 9 feet and a width of 6 feet. The height is approximately 10.5 feet. Estimate the volume of the pyramid. The estimated volume of the pyramid is 189 cubic feet.  Activity SOLUTIONS: 4. The volume of a rectangular pyramid is 450 𝑚 , and the base area is 3 75 𝑚2. Find the height of the pyramid. 1350 = ℎ 75 75 18 = ℎ 450 = 3 ℎ ℎ = 18 𝑚 75 450(3) = 3 (3)ℎ The height of the 1350 = 75ℎ pyramid is 18 meters.  Up Next Volume of a Rectangular Pyramid Solving Problems Thank You for Listening Grade 7 MATATAG Solving Problems: Volume of Rectangular Pyramid  OBJECTIVES solve problems involving the volumes of rectangular pyramids  Recap Volume of a Volume of a Rectangular Prism Rectangular Pyramid Solving Problems Involving Volume of Rectangular Pyramid  Solving Volume of a Rectangular Pyramid Example 1. A construction company is building a monument shaped like a rectangular pyramid. The base of the pyramid has an area of 150 𝑓𝑡 , and the height of the pyramid is 12 𝑓𝑡. What is 2 the volume of the monument that will be filled with concrete? Given: Solution: 2 1 𝐵 = 150 𝑓𝑡 𝑉pyramid = 𝐵ℎ 3 1 ℎ = 12 𝑓𝑡 𝑉 = (150)(12) 3 𝑉=? 𝑉= 1 (1800) 3 The volume of the monument that 𝑉 = 600 𝑓𝑡 3 will be filled with concrete is 600 𝑓𝑡 3  Solving Volume of a Rectangular Pyramid Example 2. A farmer is building a storage shed for hay, and he wants to design it in the shape of a rectangular pyramid. The base of the shed is 8 𝑚𝑒𝑡𝑒𝑟𝑠 long and 6 𝑚𝑒𝑡𝑒𝑟𝑠 wide, and the height of the shed will be 10 𝑚𝑒𝑡𝑒𝑟𝑠. How much space will be available for storing hay inside the pyramid-shaped shed? Given: Solution: 1 𝑙=8𝑚 𝑉pyramid = 𝑙𝑤ℎ 3 1 𝑤=6𝑚 𝑉 = (8)(6)(10) 3 ℎ = 10 𝑚 𝑉= 1 (480) 3 𝑉=? 3 𝑉 = 160 𝑚 The farmer can store 160 𝑚3 of hay.  Solving Volume of a Rectangular Pyramid Example 3. An architect is designing a glass pyramid for a museum. The base of the pyramid is 9 𝑓𝑡 long and 5 𝑓𝑡 wide, and the height of the pyramid will be 11.5 𝑓𝑡. Estimate the volume of the pyramid to calculate the amount of glass needed. Given: Solution: 1 𝑙 = 9 𝑓𝑡 𝑉pyramid = 𝑙𝑤ℎ 3 1 𝑤 = 5 𝑓𝑡 𝑉 = (9)(5)(11.5) 3 ℎ = 11.5 𝑓𝑡 𝑉= 1 (517.5) 3 𝑉=? 3 𝑉 = 172.5 𝑓𝑡 The estimated volume is 172.5 𝑓𝑡 3.  Solving Volume of a Rectangular Pyramid Example 4. A contractor is filling a rectangular pyramid- shaped fountain with water. The volume of the pyramid is 900 𝑚3 , and the base area is 225 𝑚2. What is the height of the pyramid-shaped fountain? 225 900 = 3 ℎ Given: Solution: 900(3) = 225 (3)ℎ 1 3 𝐵 = 225 𝑚 2 𝑉pyramid = 𝐵ℎ 3 𝑉 = 900 𝑚 3 1 2700 = 225ℎ 900 = (225)ℎ 2700 3 ℎ=? =ℎ 225 The height of the ℎ = 12 𝑚 pyramid is 12 meters.  Solving Volume of a Rectangular Pyramid Example 5. An artist is creating a sculpture shaped like a rectangular pyramid. The base of the sculpture is 7 𝑚 long, 4 𝑚 wide, and the height is 9 𝑚. How much material will the artist need to fill the pyramid? Given: Solution: 1 𝑙=7𝑚 𝑉pyramid = 𝑙𝑤ℎ 3 1 𝑤=4𝑚 𝑉 = (7)(4)(9) 3 ℎ=9𝑚 𝑉= 1 (252) 3 𝑉=? 𝑉 = 84 𝑚 3 The volume of the sculpture is 84 cubic meters.  Activity Find the volume of the solid figure with the following measurements: 1. A company is building a pyramid-shaped skylight on the roof of a mall. The base area of the skylight is 180 𝑓𝑡 2 , and the height is 10 𝑓𝑡. How much air space will be enclosed by the skylight? 2.A warehouse wants to build a pyramid-shaped container to store grain. The base of the container is 12 meters long and 7 meters wide, and the height is 15 meters. Find the volume of the container. 3. An engineer is designing a rectangular pyramid-shaped canopy for a park. The base of the canopy is 8 𝑓𝑡 long and 5 𝑓𝑡 wide, and the height is 13.2 𝑓𝑡. Estimate the volume of the canopy to calculate how much support material is needed.  Activity Find the volume of the solid figure with the following measurements: 4. A water park is constructing a pyramid-shaped water slide. The volume of the pyramid is 1,200 𝑚3 , and the base area is 300 𝑚2. Find the height of the pyramid-shaped water slide. 5. A builder is designing a pyramid-shaped roof for a house. The base of the roof is 9 𝑚𝑒𝑡𝑒𝑟𝑠 long, 5 𝑚𝑒𝑡𝑒𝑟𝑠 wide, and the height is 10 𝑚𝑒𝑡𝑒𝑟𝑠. How much space will be inside the pyramid-shaped roof?  Activity SOLUTIONS: 1. A company is building a pyramid-shaped skylight on the roof of a mall. The base area of the skylight is 180 𝑓𝑡 2 , and the height is 10 𝑓𝑡. How much air space will be enclosed by the skylight? 3 The volume of the skylight is 600 𝑓𝑡.  Activity SOLUTIONS: 2. A warehouse wants to build a pyramid-shaped container to store grain. The base of the container is 12 meters long and 7 meters wide, and the height is 15 meters. Find the volume of the container. 3 The volume of the grain container is 420 𝑚.  Activity SOLUTIONS: 3. An engineer is designing a rectangular pyramid-shaped canopy for a park. The base of the canopy is 8 𝑓𝑡 long and 5 𝑓𝑡 wide, and the height is 13.2 𝑓𝑡. Estimate the volume of the canopy to calculate how much support material is needed. The estimated volume of the canopy is 176 𝑓𝑡 3.  Activity SOLUTIONS: 4. A water park is constructing a pyramid-shaped water slide. The volume of the pyramid is 1,200 𝑚3 , and the base area is 300 𝑚2. Find the height of the pyramid-shaped water slide. The height of the water slide pyramid is 12 m.  Activity SOLUTIONS: 5. A builder is designing a pyramid-shaped roof for a house. The base of the roof is 9 𝑚𝑒𝑡𝑒𝑟𝑠 long, 5 𝑚𝑒𝑡𝑒𝑟𝑠 wide, and the height is 10 𝑚𝑒𝑡𝑒𝑟𝑠. How much space will be inside the pyramid-shaped roof? 3 The volume of the pyramid-shaped roof is 150 𝑚. Thank You for Listening Grade 7 MATATAG Volume of Square Pyramid  OBJECTIVES explain inductively the volume of a square pyramid, leading to the identification of the formula find the volume of a square pyramid  Recap Rectangular Pyramid A rectangular pyramid is a type of pyramid where the base is a rectangle. It has four triangular faces that meet at the apex, and each triangular face is connected to one side of the rectangular base.  Recap How is the volume of a rectangular pyramid derived?  Recap Volume of a Rectangular Pyramid  Volume of a Square Pyramid  Square Pyramid A Square Pyramid is a three- dimensional geometric shape consisting of four triangular sides connected at a vertex and a square base. A square foundation and four triangles joined to a vertex make up a square pyramid.  Recap Area of a Volume of a Square Rectangular Prism V=lxwxh Volume of a Square Pyramid s s  Solving Volume of a Square Pyramid  Solving Volume of a Square Pyramid Example 1. Find the volume of a square pyramid if its base edge is 9 𝑚𝑒𝑡𝑒𝑟𝑠 and its height is 14 𝑚𝑒𝑡𝑒𝑟𝑠. Solution: 1 2 𝑉pyramid = 𝑠 ℎ 3 ℎ = 14 𝑚 1 2 1 2 𝑉 = (9) (14) or 𝑉 = × (9) × (14) 3 3 1 𝑉 = 3 (81)(14) 𝑠 =9𝑚 1 𝑉 = 3 (1,134) 3 3 𝑉 = 378 𝑚 The volume of the square pyramid is 378 𝑚.  Solving Volume of a Square Pyramid Example 2. Find the volume of a square pyramid if its height is 12 𝑖𝑛 and its base edge is 7 𝑖𝑛. Solution: 1 2 𝑉pyramid = 𝑠 ℎ 3 ℎ = 12 𝑖𝑛 1 2 𝑉 = (7) (12) 3 1 𝑉 = 3 (49)(12) 1 𝑠 = 7 𝑖𝑛 𝑉 = 3 (588) 3 The volume of the square pyramid is 196 𝑖𝑛3. 𝑉 = 196 𝑖𝑛  Solving Volume of a Square Pyramid Example 3. A square pyramid has a base area of 81 and 𝑓𝑡 2 a height of 7 𝑓𝑡. Determine the volume of the pyramid. Solution: 1 𝑉pyramid = 𝐵ℎ 3 1 𝑉 = (81)(7) ℎ = 7 𝑓𝑡 3 1 𝑉= 3 (567) 3 𝑉= 189 𝑓𝑡 The volume of the square 2 pyramid is 189 𝑓𝑡 3 𝐴 = 81 𝑓𝑡  Solving Volume of a Square Pyramid Example 4 The roof of a house is in the form of a square pyramid. If its base edge is 6 𝑓𝑒𝑒𝑡 and the volume is 120 𝑐𝑢𝑏𝑖𝑐 𝑓𝑒𝑒𝑡, how high is the roof? Given: 120 = 36 ℎ 3 3 V = 120 𝑓𝑡 𝑠 = 6 𝑓𝑡 120 = 12ℎ Solution: 120 1 2 =ℎ 𝑉pyramid = 𝑠 ℎ 12 3 1 2 ℎ = 10 𝑓𝑡 120 = (6) ℎ 3 The height of the 1 rectangular 120 = (36)ℎ pyramid is 10 ft 3  Activity Find the volume of the solid figure with the following measurements: 1. Find the volume of a square pyramid if its base edge is 6 𝑚𝑒𝑡𝑒𝑟𝑠 and its height is 12 𝑚𝑒𝑡𝑒𝑟𝑠. 2.Find the volume of a square pyramid if its height is 15 𝑖𝑛𝑐ℎ𝑒𝑠 and its base edge is 10 𝑖𝑛𝑐ℎ𝑒𝑠. 3. A square pyramid has a base area of 64 𝑓𝑡 2 and a height of 9 𝑓𝑡. Determine the volume of the pyramid. 4. The roof of a house is in the form of a square pyramid. If its base edge is 5 𝑓𝑒𝑒𝑡 and the volume is 75 𝑐𝑢𝑏𝑖𝑐 𝑓𝑒𝑒𝑡, how high is the roof?  Activity SOLUTIONS: 1. Find the volume of a square pyramid if its base edge is 6 𝑚𝑒𝑡𝑒𝑟𝑠 and its height is 12 𝑚𝑒𝑡𝑒𝑟𝑠. The volume of the square pyramid is 144 m³.  Activity SOLUTIONS: 2. Find the volume of a square pyramid if its height is 15 𝑖𝑛𝑐ℎ𝑒𝑠 and its base edge is 10 𝑖𝑛𝑐ℎ𝑒𝑠. The volume of the square pyramid is 500 in³.  Activity SOLUTIONS: 3. A square pyramid has a base area of 64 𝑓𝑡 and a height of 9 𝑓𝑡. 2 Determine the volume of the pyramid. The volume of the square 3 pyramid is 192 𝑓𝑡  Activity SOLUTIONS: 4. The roof of a house is in the form of a square pyramid. If its base edge is 5 𝑓𝑒𝑒𝑡 and the volume is 75 𝑐𝑢𝑏𝑖𝑐 𝑓𝑒𝑒𝑡, how high is the roof? The height of the pyramid is 9 feet.  Up Next Volume of a Square Pyramid Solving Problems Thank You for Listening Grade 7 MATATAG Solving Problems: Volume of Square Pyramid  OBJECTIVES solve problems involving the volumes of square pyramids  Recap Volume of a Volume of a Rectangular Prism Square Pyramid s Solving Problems Involving Volume of Square Pyramid  Solving Volume of a Square Pyramid Example 1. A gardener is planning to build a decorative square pyramid-shaped planter in his backyard. If the base edge of the planter will be 5 𝑚𝑒𝑡𝑒𝑟𝑠 and its height will be 9 𝑚𝑒𝑡𝑒𝑟𝑠, what will be the volume of the planter to determine how much soil he needs to fill it? Solution: 1 2 𝑉pyramid = 𝑠 ℎ 3 3 𝑉= 75 𝑚 ℎ=9𝑚 1 2 𝑉 = (5) (9) The volume of the 3 square pyramid is 1 75 𝑚3. 𝑉 = 3 (25)(9) 𝑠 =5𝑚 1 𝑉 = 3 (225)  Solving Volume of a Square Pyramid Example 2. A museum is designing a display case in the shape of a square pyramid. The base of the pyramid has an area of 49 𝑚 ,2 and the height of the pyramid is 15 𝑚. What is the volume of the display case? Solution: ℎ = 15 𝑚 1 𝑉pyramid = 𝐵ℎ 3 1 𝑉 = (49)(15) 3 1 2 𝑉= (735) 𝐴 = 49 𝑚 3 3 The volume of the square pyramid is 245 𝑚3 𝑉 = 245 𝑚  Solving Volume of a Square Pyramid Example 3. A resort in Palawan is designing a square pyramid- shaped lookout tower to offer tourists a panoramic view of the island. The base of the tower has an area of 100 𝑠𝑞𝑢𝑎𝑟𝑒 𝑚𝑒𝑡𝑒𝑟𝑠, and the volume of the tower will be 300 𝑐𝑢𝑏𝑖𝑐 𝑚𝑒𝑡𝑒𝑟𝑠. What is the height of the tower? Given: 100 300 = 3 ℎ 3 V = 300 𝑚 100 B = 100 𝑚 300(3) = 3 (3)ℎ Solution: 1 900 = 100ℎ ℎ =9𝑚 𝑉pyramid = 𝐵ℎ 3 900 The height of the lookout tower 1 300 = (100)ℎ = ℎ 100 is 9 meters. 3  Solving Volume of a Square Pyramid Example 4. The composite solid is composed of a rectangular prism and a square pyramid on top. To find its volume, we need to add the volumes of the prism and the pyramid. 1 2 Solution: 𝑉pyramid = 𝑠 ℎ 3 𝑉prism = 𝐵ℎ 1 2 𝑉pyramid = 4 (6) 𝑉prism = 𝑙𝑤ℎ 3 1 𝑉prism = (4)(4)(5) 𝑉pyramid = (96) 3 3 𝑉pyramid = 32 𝑖𝑛 3 𝑉prism = 80 𝑖𝑛 𝑉total = 80 𝑖𝑛 3 + 32 𝑖𝑛 3 3 = 112 𝑖𝑛 The volume of the composite solid is 112 𝑖𝑛3.  Activity Find the volume of the solid figure with the following measurements: 1. A landscaper is building a decorative square pyramid-shaped fountain in a city park. The base edge of the fountain is 6 𝑚, and its height is 12 𝑚. What will be the volume of the fountain to determine how much water it can hold? 2. A company is creating a square pyramid-shaped monument for a town plaza. The base area of the monument is 64 𝑚2 , and its height is 18 𝑚. What is the volume of the monument? 3. An art installation in Baguio City will feature a square pyramid-shaped sculpture. The base of the pyramid has an area of 81 𝑚2 , and the total volume of the sculpture will be 405 𝑚3. How tall will the sculpture be? 4. A designer is creating a decorative structure that consists of a rectangular prism at the base and a square pyramid on top. The rectangular prism has a base area of 50 𝑚2 and a height of 8 𝑚. The square pyramid has a base edge of 5 𝑚 and a height of 6 𝑚. What is the total volume of the structure?  Activity SOLUTIONS: 1. A landscaper is building a decorative square pyramid-shaped fountain in a city park. The base edge of the fountain is 6 𝑚, and its height is 12 𝑚. What will be the volume of the fountain to determine how much water it can hold?  Activity SOLUTIONS: 2. A company is creating a square pyramid-shaped monument for a town plaza. The base area of the monument is 64 𝑚2 , and its height is 18 𝑚. What is the volume of the monument?  Activity SOLUTIONS: 3. An art installation in Baguio City will feature a square pyramid- shaped sculpture. The base of the pyramid has an area of 81 𝑚2 , and the total volume of the sculpture will be 405 𝑚3. How tall will the sculpture be?  Activity SOLUTIONS: 4. A designer is creating a decorative structure that consists of a rectangular prism at the base and a square pyramid on top. The rectangular prism has a base area of 50 𝑚2 and a height of 8 𝑚. The square pyramid has a base edge of 5 𝑚 and a height of 6 𝑚. What is the total volume of the structure? Thank You for Listening