Lesson 8.4 Regular Polygons pp Lesson 8.4 Regular Polygons pp

Objectives: 1.To define terms related to regular polygons. 2.To derive and apply a formula for the area of regular polygons. 3.To find relationships between the apothem, altitude, and side of an equilateral triangle. Objectives: 1.To define terms related to regular polygons. 2.To derive and apply a formula for the area of regular polygons. 3.To find relationships between the apothem, altitude, and side of an equilateral triangle.

The center of a regular polygon is the common center of the inscribed and circumscribed circles of the regular polygon. DefinitionDefinition

The radius of a regular polygon is a segment that joins the center of a regular polygon with one of its vertices. DefinitionDefinition

The apothem of a regular polygon is the perpendicular segment that joins the center with a side of the polygon. DefinitionDefinition

a a r r C C The special parts of a regular polygon

A central angle of a regular polygon is the angle formed at the center of the polygon by two radii drawn to consecutive vertices. DefinitionDefinition

P P R R Q Q  RPQ is a central angle.

Theorem 8.9 The central angles of a regular n -gon are congruent and measure. Theorem 8.9 The central angles of a regular n -gon are congruent and measure. 360° n 360° n

EXAMPLE 1 What are the measures of the central angles and the base angles of the regular octagon? P R Q m  RPQ = = 45° 360° 8 360° 8 180° – 45° = 135° m  PRQ = m  PQR = = 67.5° 135° 2 135° 2

Theorem 8.10 The area of a regular polygon is one- half the product of its apothem and its perimeter: A = ap. Theorem 8.10 The area of a regular polygon is one- half the product of its apothem and its perimeter: A = ap

Theorem 8.11 The apothem of an equilateral triangle is one-third the length of the altitude: a = h. Theorem 8.11 The apothem of an equilateral triangle is one-third the length of the altitude: a = h

Theorem 8.12 The apothem of an equilateral triangle is 3 times one-sixth the length of the side: a = s. Theorem 8.12 The apothem of an equilateral triangle is 3 times one-sixth the length of the side: a = s

EXAMPLE 2 Find the area of the following regular octagon p = ns = 8(10) = 80 A = ap A = 480 A = (12)(80)

EXAMPLE 3 Find the area of the following regular hexagon. a a Q Q P P C C 4 4 D D Central angle = 360° 6 360° 6 Central angle = 60° Base angles = 120° 2 120° 2 Base angle = 60° ∆CPQ is equiangular & equilateral

C C P P D D Q Q a a a = 4 2 a = 16 a 2 = 12 a = 12 a = 2 3

EXAMPLE 3 Find the area of the following regular hexagon. a a Q Q P P C C 4 4 D D A = ap A = (2 3)(24) A = 24 3 A ≈ 41.6 sq. units

Homework pp Homework pp

►A. Exercises 5.Find the area. ►A. Exercises 5.Find the area. A = ½ap A = ½(29)(192) A = 2784 sq. units A = ½ap A = ½(29)(192) A = 2784 sq. units 24 29

►A. Exercises 8.Find the area. ►A. Exercises 8.Find the area

No. of sides of a reg. poly. Length of Apothem (units) Length of side (units) Length of radius (units) Area (sq. units) ►B. Exercises Complete the following table. ►B. Exercises Complete the following table

No. of sides of a reg. poly. Length of Apothem (units) Length of side (units) Length of radius (units) Area (sq. units) ►B. Exercises Complete the following table. ►B. Exercises Complete the following table

►B. Exercises Prove the formulas in exercises However, use derivations with formulas rather than two-column proofs. 14.Prove Theorem 8.10 for the case of a regular quadrilateral by showing that the formula for the area of a regular quadrilateral reduces to the formula for the area of a square. ►B. Exercises Prove the formulas in exercises However, use derivations with formulas rather than two-column proofs. 14.Prove Theorem 8.10 for the case of a regular quadrilateral by showing that the formula for the area of a regular quadrilateral reduces to the formula for the area of a square.

►B. Exercises Prove the formulas in exercises However, use derivations with formulas rather than two-column proofs. 14. ►B. Exercises Prove the formulas in exercises However, use derivations with formulas rather than two-column proofs. 14. s s s s ½s

►B. Exercises Prove the formulas in exercises However, use derivations with formulas rather than two-column proofs. 15.Theorem Use the two formulas for the area of a triangle to show that a = for an equilateral triangle. ►B. Exercises Prove the formulas in exercises However, use derivations with formulas rather than two-column proofs. 15.Theorem Use the two formulas for the area of a triangle to show that a = for an equilateral triangle. h3h3 h3h3

■ Cumulative Review Give the name of each. 21.A cylinder with polygonal bases ■ Cumulative Review Give the name of each. 21.A cylinder with polygonal bases

■ Cumulative Review Give the name of each. 22.A cone with a polygonal base ■ Cumulative Review Give the name of each. 22.A cone with a polygonal base

■ Cumulative Review Give the name of each. 23.A pair of lines that are neither parallel nor intersecting ■ Cumulative Review Give the name of each. 23.A pair of lines that are neither parallel nor intersecting

■ Cumulative Review Give the name of each. 24.The form of the argument shown p  q p Therefore q. ■ Cumulative Review Give the name of each. 24.The form of the argument shown p  q p Therefore q.

■ Cumulative Review Give the name of each. 25.Adjacent angles in which the noncommon sides form opposite rays ■ Cumulative Review Give the name of each. 25.Adjacent angles in which the noncommon sides form opposite rays