Geometry: Perimeter and Area

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Questions and Answers

What is the formula for calculating the perimeter of a rectangle?

The perimeter of a rectangle is calculated using the formula $P = 2 \times (length + breadth)$.

How do you find the perimeter of a square?

The perimeter of a square is found by using the formula $P = 4 \times length$ of its side.

If a triangle has sides measuring 4 cm, 5 cm, and 7 cm, what is its perimeter?

The perimeter of the triangle is $4 cm + 5 cm + 7 cm = 16 cm$.

What would be the length of lace needed to go around a rectangular tablecloth that is 3 m long and 2 m wide?

<p>The length of lace required is $P = 2 \times (3 m + 2 m) = 10 m$.</p> Signup and view all the answers

Explain the symmetry property of a rectangle's sides regarding perimeter calculation.

<p>In a rectangle, opposite sides are equal, so the perimeter can be calculated as the sum of two lengths and two widths.</p> Signup and view all the answers

How is the calculation for the perimeter of a square different from that of a rectangle?

<p>The perimeter of a square uses all equal sides, calculated as $P = 4 \times length$, while a rectangle considers both length and width.</p> Signup and view all the answers

Why is it sufficient to multiply the length of one side by 4 to find the perimeter of a square?

<p>All four sides of a square are identical in length, making multiplication by 4 a quick calculation for the perimeter.</p> Signup and view all the answers

What is the total perimeter of a rectangle with a length of 12 cm and a breadth of 8 cm?

<p>The total perimeter is $40 cm$ calculated as $2 \times (12 cm + 8 cm)$.</p> Signup and view all the answers

What is the area of a rectangle with length 5 m and width 4 m?

<p>20 sq m</p> Signup and view all the answers

In a rectangular garden that is 25 m long with an area of 300 sq m, what is the width?

<p>12 m</p> Signup and view all the answers

What is the maximum number of coconut trees that can be planted in a 100 m by 50 m grove if each tree requires 25 sq m?

<p>200 trees</p> Signup and view all the answers

Calculate the perimeter of the arrangement with dimensions 6 cm and height 2 cm.

<p>16 cm</p> Signup and view all the answers

What is the area of the floor that is not covered by a 3 m square carpet on a 5 m by 4 m floor?

<p>11 sq m</p> Signup and view all the answers

If a rectangular plot is 500 m long and 200 m wide, what is the cost of tiling it at ` 8 per hundred sq m?

<p>` 80</p> Signup and view all the answers

How do you find the area of irregular shapes using squared paper?

<p>Trace the shape and count the full squares while applying conventions.</p> Signup and view all the answers

What is the area of four square flower beds each with a side length of 4 m on a plot 12 m by 10 m?

<p>56 sq m</p> Signup and view all the answers

What shape has more area: a square with side 3 m or a rectangle with dimensions 3 m x 4 m?

<p>Rectangle has more area.</p> Signup and view all the answers

In terms of area, how does Shape D relate to Shape C?

<p>Shape D has twice the area of Shape C.</p> Signup and view all the answers

What shape arrangement provides a perimeter of 22 cm using two pieces cut from a rectangle?

<p>Two pieces arranged to form a rectangle with a specific measurement.</p> Signup and view all the answers

Using the dimensions 6 cm and 4 cm, what is the perimeter of a rectangle?

<p>20 cm</p> Signup and view all the answers

What is the main reason squares are used to measure area?

<p>Squares fit perfectly without gaps and make area calculation straightforward.</p> Signup and view all the answers

What is the width of a rectangular garden that has a long side of 50 m and an area of 1000 sq m?

<p>20 m</p> Signup and view all the answers

Calculate the area of the floor that is not carpeted when a square carpet of side 3 m is placed in a room that is 5 m by 4 m.

<p>14 sq m</p> Signup and view all the answers

After digging four flower beds at the corners of a garden measuring 15 m by 12 m, what area remains for the lawn?

<p>132 sq m</p> Signup and view all the answers

If shapes A and B have areas of 18 and 20 square units respectively, can A have a longer perimeter than B? Give an example of such shapes.

<p>Yes, a long rectangle for A and a compact shape for B can satisfy this.</p> Signup and view all the answers

What is the perimeter of a rectangular border that is 1 cm from the top and bottom and 1.5 cm from the left and right sides of a page?

<p>Perimeter = 2(Length + Width) where Length = page length - 3 cm and Width = page width - 2 cm.</p> Signup and view all the answers

If you draw a rectangle measuring 12 units by 8 units and then another rectangle inside it that occupies exactly half the area, what will be the area of the smaller rectangle?

<p>48 sq units</p> Signup and view all the answers

When a square piece of paper is folded in half and cut into two rectangles, which statement about area and perimeter is always true?

<p>The perimeter of the square is greater than the combined perimeters of the two rectangles.</p> Signup and view all the answers

Can two different shapes have the same area but different perimeters? Give an example if so.

<p>Yes, a square and a rectangle can have the same area but different perimeters.</p> Signup and view all the answers

What are the dimensions of the rectangle that has the greatest perimeter with an area of 24 square units?

<p>6 units by 4 units.</p> Signup and view all the answers

Which rectangle with an area of 24 square units has the least perimeter?

<p>4 units by 6 units.</p> Signup and view all the answers

Why can’t we use circles instead of squares to find the area on grid paper?

<p>Circles do not align with the grid and their area cannot be expressed in whole units with the square grid.</p> Signup and view all the answers

What would be the dimensions of a rectangle with an area of 32 sq cm that has the greatest perimeter?

<p>8 units by 4 units.</p> Signup and view all the answers

Is it possible to predict the shape of the rectangle with the greatest perimeter given any area? Provide reasons.

<p>Yes, as the rectangle approaches a square shape, the perimeter decreases.</p> Signup and view all the answers

When examining two figures made with 9 unit squares, what is the smallest possible perimeter?

<p>12 units.</p> Signup and view all the answers

What is the maximum perimeter you can achieve with 9 unit squares?

<p>20 units.</p> Signup and view all the answers

If you attach a new square to an existing figure of perimeter 24 units, what will happen to the perimeter?

<p>It could either increase, decrease, or remain the same depending on the attachment.</p> Signup and view all the answers

What occurs to the areas of the two triangles formed by cutting a rectangle along its diagonal?

<p>The two triangles have equal areas.</p> Signup and view all the answers

What is the relationship between the area of a rectangle and the area of triangles formed from it?

<p>The area of a triangle is always half the area of the rectangle it comes from.</p> Signup and view all the answers

How does the area of triangle ABE relate to the areas of rectangles AFED and BFEC?

<p>Area of triangle ABE equals half the area of rectangle ABCD.</p> Signup and view all the answers

What are the missing dimensions of Charan’s hall if the area is to be determined?

<p>The dimensions need to be calculated using the known values.</p> Signup and view all the answers

What is the total area of Sharan’s kitchen if the dimensions are 18 ft by 10 ft?

<p>180 sq ft.</p> Signup and view all the answers

Which figure arrangement with 9 unit squares has a perimeter of 18 units?

<p>One possible arrangement is a 3x1 and a 2x3 figure combined.</p> Signup and view all the answers

What is the length of lace required for a rectangular table cover that has a length of 3 m and a breadth of 2 m?

<p>10 m</p> Signup and view all the answers

If Usha takes three rounds of a square park with each side measuring 75 m, what is the total distance she travels?

<p>900 m</p> Signup and view all the answers

Given a rectangle with a perimeter of 14 cm and a breadth of 2 cm, what is the length?

<p>6 cm</p> Signup and view all the answers

What is the side length of a square formed from a wire that originally outlined a rectangle with dimensions 5 cm by 3 cm?

<p>2 cm</p> Signup and view all the answers

If a triangle has a perimeter of 55 cm and two sides measuring 20 cm and 14 cm, what is the length of the third side?

<p>21 cm</p> Signup and view all the answers

What is the cost of fencing a rectangular park with a length of 150 m and breadth of 120 m, if the cost is $40 per meter?

<p>$10,800</p> Signup and view all the answers

With a 36 cm string, how long will each side be if it is formed into an equilateral triangle?

<p>12 cm</p> Signup and view all the answers

What is the total length of rope needed for a rectangular field with a length of 230 m, breadth of 160 m, when using 3 rounds of rope?

<p>1,380 m</p> Signup and view all the answers

How far does Akshi run in total if she completes 5 rounds on a rectangular track measuring 70 m by 40 m?

<p>1,100 m</p> Signup and view all the answers

What is the total distance Toshi covers in 7 rounds along a similar rectangular track measuring 70 m by 40 m?

<p>1,540 m</p> Signup and view all the answers

If an equilateral triangle has a side length of 3 m, what is its perimeter?

<p>9 m</p> Signup and view all the answers

For a regular hexagon formed from a string of length 48 cm, what is the length of each side?

<p>8 cm</p> Signup and view all the answers

If two square running tracks each have a side of 100 m and 150 m respectively, what is the perimeter of the larger track?

<p>600 m</p> Signup and view all the answers

What is the total distance if a runner completes 3 rounds on a square track with side length 100 m?

<p>1,200 m</p> Signup and view all the answers

For a shape with a red length of 6 units and a blue diagonal length of 3 units, what is the expressed perimeter?

<p>6s + 3d units</p> Signup and view all the answers

Flashcards

What is the perimeter?

The total distance around the outer boundary of a closed plane figure.

How to calculate the perimeter of a polygon?

The sum of the lengths of all sides of a polygon.

What is the formula for the perimeter of a rectangle?

Twice the sum of its length and breadth.

How to calculate the perimeter of a square?

Quadruple the length of one of its sides.

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How to find the perimeter of a triangle?

The sum of the lengths of its three sides.

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Perimeter

The distance around the outer edge of a shape.

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What is a polygon?

A closed plane figure made up of line segments.

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What is a rectangle?

A closed plane figure with four sides and four right angles.

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Area

The amount of space a two-dimensional shape covers.

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Rectangle

A rectangle is a four-sided shape with four right angles and opposite sides equal.

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Square

A square is a four-sided shape with four right angles and all sides equal.

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Perimeter of a rectangle

The formula for finding the perimeter of a rectangle is 2 * (length + width).

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Area of a rectangle

The formula for finding the area of a rectangle is length * width.

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Perimeter of a square

The formula for finding the perimeter of a square is 4 * side.

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Area of a square

The formula for finding the area of a square is side * side.

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Triangle

A shape with three sides and three angles.

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Perimeter of a triangle

The sum of the lengths of all three sides of a triangle.

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Regular polygon

A closed shape with all sides and all angles equal.

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Equilateral triangle

A triangle with all three sides equal in length.

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Perimeter of an equilateral triangle

The perimeter of an equilateral triangle is three times the length of one side.

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One round

The distance covered by moving along the boundary of a shape once.

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Multiple rounds

The total distance covered by walking around the entire perimeter of a shape multiple times.

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Calculating perimeter

A way to find out the perimeter of a shape by adding the lengths of all its sides.

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Measuring perimeter

The act of using a scale or measuring tape to determine the actual perimeter of a shape.

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Estimating perimeter

The process of estimating a perimeter without accurate measurement tools.

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Square as a regular polygon

A special case of a regular polygon where all sides are equal and all angles are right angles.

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Diagonal

A line segment that connects two opposite corners of a rectangle.

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Area of a triangle

Half the area of the rectangle it is formed from.

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Polygon

A closed figure with a fixed number of sides and angles.

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Rectangle with least perimeter

The perimeter of a rectangle with the greatest area for a given area.

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Rectangle with greatest perimeter

The perimeter of a rectangle with the least area for a given area.

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Area of composite shapes

Finding the total area of shapes by dividing them into simpler shapes.

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Area on a grid paper

Calculating the area of squares and rectangles by counting the number of unit squares.

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House plan

A plan of a house or building.

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Layout of a house

The arrangement of rooms and their sizes in a house plan.

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Problem-solving skills

The ability to analyze and solve area and perimeter problems in various contexts.

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Splitting a figure

The process of dividing a figure into smaller parts.

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Joining figures

The process of combining two or more figures to create a new figure.

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Tracing a shape on squared paper

The act of tracing the outline of a figure on squared paper to estimate its area.

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Area estimation conventions on squared paper

The convention used to count squares when estimating area on squared paper.

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Estimating area

The ability to estimate the area of a figure using squared paper.

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Why squares are best for measuring area

The reason why squares are the best units for measuring area.

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Circle

A shape with a circular boundary.

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Diameter

The distance across a circle through its center.

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Coconut grove

A part of a land where trees are cultivated.

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Study Notes

Perimeter

  • Perimeter is the total distance around the boundary of a closed plane figure.
  • For polygons, perimeter is the sum of the lengths of all sides.
  • Perimeter of a rectangle = 2 × (length + width)
  • Perimeter of a square = 4 × side length
  • Perimeter of a triangle = sum of the lengths of all three sides

Area

  • Area is the amount of space enclosed by a closed figure.
  • Area is typically measured in square units.
  • Area of a rectangle = length × width
  • Area of a square = side × side
  • Area of a triangle = ½ × base × height

Relationships Between Perimeter and Area

  • Figures can have the same area but different perimeters, or the same perimeter but different areas.
  • Area can be estimated by dividing a figure into squares or rectangles and triangles.

Specific Examples and Calculations

  • Real-world examples illustrate calculating perimeter and area for rectangles, squares, and triangles, along with compound shapes (e.g., a rectangular floor with a flowerbed).
  • Calculating the area of figures broken down into smaller rectangles and triangles.
  • Estimating the area of irregular shapes using grid paper.
  • Problem-solving examples include finding missing dimensions, calculating costs of fencing, and determining the length of lace needed.
  • Examples for comparing lengths of running tracks and finding strategies to achieve a common finish line for runners by running a prescribed distance.
  • Examples on calculating perimeter for shapes with both straight and "diagonal" lines.

Tangram Puzzles

  • Tangram pieces can be used to explore relationships between areas of different shapes.

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