Pentagon Shapes and Properties

Polygons

• A pentagon is a polygon with 5 straight sides and 5 angles.
• There are different types of pentagons, including:
  • Regular pentagon: all sides and angles are equal.
  • Irregular pentagon: not all sides or angles are equal.
  • Convex pentagon: all sides point out away from the pentagon.
  • Concave pentagon: one side points in toward the pentagon.
  • Equilateral pentagon: all side lengths are equal but not all angles are.
  • Cyclic pentagon: if circumscribed, all points would fall along the circle.
• The area of a regular pentagon can be calculated using the formula: Area = (5s^2)/(4a), where s is the length of the side and a is the length of the apothem.
• The perimeter of a regular or equilateral pentagon can be calculated using the formula: Perimeter = 5s, where s is the length of one side.
• Pentagons can be found in everyday items, such as soccer balls, home plates, and crosswalk signs.

Hexagons

• A hexagon is a closed geometrical shape with six sides and six angles.
• There are different types of hexagons, including:
  • Regular hexagon: equal sides and equal angles.
  • Irregular hexagon: varied measurements of sides and angles.
  • Convex hexagon: all interior angles are less than 180 degrees.
  • Concave hexagon: at least one interior angle is greater than 180 degrees.
  • Complex hexagons: formed by the intersection of different shapes, differing from regular hexagons in structure and formation.
• Hexagons can be found in nature, such as in honeycombs, snowflakes, rocks, and minerals, and in the center scutes on a turtle's shell.

Octagons

• An octagon is an eight-sided polygon.
• There are four types of octagons, including:
  • Regular octagon: congruent interior angles (all 135 degrees), congruent sides, and congruent exterior angles (all 45 degrees).
  • Irregular octagon: varied measurements of sides and angles.
  • Convex octagon: all interior angles are less than 180 degrees.
  • Concave octagon: at least one interior angle is greater than 180 degrees.
• The total interior angle measure of an octagon is 1080 degrees.
• The equations for an octagon can be predicted based on its properties.

Measurement Systems

• The imperial system measures weight, volume, and distance in pounds, gallons, and feet, respectively.
• The metric system measures weight, volume, and distance in grams, liters, and meters, respectively.
• The metric system has base units for multiple types of measurements, including:
  • Mass: gram (g)
  • Distance: meter (m)
  • Time: second (s)
  • Amount of a chemical substance: mole (mol)
  • Temperature: degree Celsius (C)
  • Electrical current: ampere (A)
  • Light intensity: candela (cd)
  • Volume: liter (L)
• Metric measurements can be converted to larger or smaller units by multiplying or dividing by a power of ten.
• The metric system is the most commonly used system of measurement around the world.

Converting Units

• When converting units in the metric system, remember to:
  • Identify what you're starting with.
  • Identify what you need to get to.
  • Use the conversion factor to get from the starting unit to the desired unit.
• Always place the units of what you're starting with on the opposite side of the fraction line from those same units in the conversion factor.

Time and Its Measurement

• The International System of Units (SI) defines the second as the unit of time.
• Other commonly used units of time include minutes, hours, days, weeks, and months.
• Time and its measurement study is called horology.
• Astronomical objects were used to determine time, but standard units and clocks made it easier to measure time.
• Important relationships between units include:
  • 1 minute = 60 seconds
  • 1 hour = 60 minutes
  • 1 hour = 3600 seconds
  • 1 day = 24 hours
  • 1 week = 7 days
• To convert a larger unit to a smaller unit, multiply; to convert a smaller unit to a larger unit, divide.

Distance and Its Measurement

• Distance can be calculated using the distance equation or the distance rate time formula.
• The distance equation is used for coordinates in the coordinate plane.
• The distance rate time formula is used for distance word problems and requires keeping the units of the rate and time the same.
• The formula can be rearranged to solve for rate or time if needed.

Mass and Volume

• Mass and volume are physical properties that quantify the amount of matter and the space occupied by matter.
• The standard units of mass and volume are kilograms and cubic meters, respectively.
• Mass and volume are extensive properties, meaning they depend on the quantity of matter present.
• The main difference between mass and volume is that mass quantifies the amount of substance, while volume measures the amount of space occupied.
• Density is an intensive property, meaning it depends on the type of matter, not its quantity.
• The formula for density is: Density = Mass / Volume

Standard Units and Accuracy

• Having standard units is important in science to ensure accurate and clear measurements.
• Rounding units can lead to inaccurate measurements.
• Standard units allow scientists to communicate effectively and efficiently.

Significant Digits and Accuracy

• Significant digits, also called significant figures or sig figs, are the number of digits used to express a calculated or measured value.
• Accuracy refers to how close the measured value is to the true or accepted value.
• Precision refers to how close measurements are to each other.
• The number of significant figures used in reporting measured values should not be more precise than the instrument used to make the measurement.
• Rules for significant figure calculations include:
  • Any non-zero digit is significant.
  • Zeros are significant if they appear between non-zero digits.
  • A zero is not significant if it comes before a decimal point.
  • Any zeros that follow a non-zero digit and come after a decimal point are significant.
  • Trailing zeros are only significant if a decimal follows the zero.

Accuracy and Percent Error

• Accuracy is defined as how close a measurement is to the true or accepted value.
• The amount of error in a measurement can be quantified by calculating the percent error between the measurement and the true or accepted value.

Polygons

• A pentagon is a polygon with 5 straight sides and 5 angles.
• There are different types of pentagons, including:
  • Regular pentagon: all sides and angles are equal.
  • Irregular pentagon: not all sides or angles are equal.
  • Convex pentagon: all sides point out away from the pentagon.
  • Concave pentagon: one side points in toward the pentagon.
  • Equilateral pentagon: all side lengths are equal but not all angles are.
  • Cyclic pentagon: if circumscribed, all points would fall along the circle.
• The area of a regular pentagon can be calculated using the formula: Area = (5s^2)/(4a), where s is the length of the side and a is the length of the apothem.
• The perimeter of a regular or equilateral pentagon can be calculated using the formula: Perimeter = 5s, where s is the length of one side.
• Pentagons can be found in everyday items, such as soccer balls, home plates, and crosswalk signs.

Hexagons

• A hexagon is a closed geometrical shape with six sides and six angles.
• There are different types of hexagons, including:
  • Regular hexagon: equal sides and equal angles.
  • Irregular hexagon: varied measurements of sides and angles.
  • Convex hexagon: all interior angles are less than 180 degrees.
  • Concave hexagon: at least one interior angle is greater than 180 degrees.
  • Complex hexagons: formed by the intersection of different shapes, differing from regular hexagons in structure and formation.
• Hexagons can be found in nature, such as in honeycombs, snowflakes, rocks, and minerals, and in the center scutes on a turtle's shell.

Octagons

• An octagon is an eight-sided polygon.
• There are four types of octagons, including:
  • Regular octagon: congruent interior angles (all 135 degrees), congruent sides, and congruent exterior angles (all 45 degrees).
  • Irregular octagon: varied measurements of sides and angles.
  • Convex octagon: all interior angles are less than 180 degrees.
  • Concave octagon: at least one interior angle is greater than 180 degrees.
• The total interior angle measure of an octagon is 1080 degrees.
• The equations for an octagon can be predicted based on its properties.

Measurement Systems

• The imperial system measures weight, volume, and distance in pounds, gallons, and feet, respectively.
• The metric system measures weight, volume, and distance in grams, liters, and meters, respectively.
• The metric system has base units for multiple types of measurements, including:
  • Mass: gram (g)
  • Distance: meter (m)
  • Time: second (s)
  • Amount of a chemical substance: mole (mol)
  • Temperature: degree Celsius (C)
  • Electrical current: ampere (A)
  • Light intensity: candela (cd)
  • Volume: liter (L)
• Metric measurements can be converted to larger or smaller units by multiplying or dividing by a power of ten.
• The metric system is the most commonly used system of measurement around the world.

Converting Units

• When converting units in the metric system, remember to:
  • Identify what you're starting with.
  • Identify what you need to get to.
  • Use the conversion factor to get from the starting unit to the desired unit.
• Always place the units of what you're starting with on the opposite side of the fraction line from those same units in the conversion factor.

Time and Its Measurement

• The International System of Units (SI) defines the second as the unit of time.
• Other commonly used units of time include minutes, hours, days, weeks, and months.
• Time and its measurement study is called horology.
• Astronomical objects were used to determine time, but standard units and clocks made it easier to measure time.
• Important relationships between units include:
  • 1 minute = 60 seconds
  • 1 hour = 60 minutes
  • 1 hour = 3600 seconds
  • 1 day = 24 hours
  • 1 week = 7 days
• To convert a larger unit to a smaller unit, multiply; to convert a smaller unit to a larger unit, divide.

Distance and Its Measurement

• Distance can be calculated using the distance equation or the distance rate time formula.
• The distance equation is used for coordinates in the coordinate plane.
• The distance rate time formula is used for distance word problems and requires keeping the units of the rate and time the same.
• The formula can be rearranged to solve for rate or time if needed.

Mass and Volume

• Mass and volume are physical properties that quantify the amount of matter and the space occupied by matter.
• The standard units of mass and volume are kilograms and cubic meters, respectively.
• Mass and volume are extensive properties, meaning they depend on the quantity of matter present.
• The main difference between mass and volume is that mass quantifies the amount of substance, while volume measures the amount of space occupied.
• Density is an intensive property, meaning it depends on the type of matter, not its quantity.
• The formula for density is: Density = Mass / Volume

Standard Units and Accuracy

• Having standard units is important in science to ensure accurate and clear measurements.
• Rounding units can lead to inaccurate measurements.
• Standard units allow scientists to communicate effectively and efficiently.

Significant Digits and Accuracy

• Significant digits, also called significant figures or sig figs, are the number of digits used to express a calculated or measured value.
• Accuracy refers to how close the measured value is to the true or accepted value.
• Precision refers to how close measurements are to each other.
• The number of significant figures used in reporting measured values should not be more precise than the instrument used to make the measurement.
• Rules for significant figure calculations include:
  • Any non-zero digit is significant.
  • Zeros are significant if they appear between non-zero digits.
  • A zero is not significant if it comes before a decimal point.
  • Any zeros that follow a non-zero digit and come after a decimal point are significant.
  • Trailing zeros are only significant if a decimal follows the zero.

Accuracy and Percent Error

• Accuracy is defined as how close a measurement is to the true or accepted value.
• The amount of error in a measurement can be quantified by calculating the percent error between the measurement and the true or accepted value.

Learn about different types of pentagons, including regular, irregular, convex, concave, and cyclic pentagons, and their properties.