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m 1 = = 60 Divide 360 by the number of sides.

360 6. m 1 = = 60 Divide 360 by the number of sides. 1 2. m 2 = ( 60 ) = 30 Substitute 60 for m 1. m 2 = m 1 The apothem bisects the vertex angle of the isosceles triangle formed by the radii. 1 2.

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m 1 = = 60 Divide 360 by the number of sides.

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  1. 360 6 m 1 = = 60 Divide 360 by the number of sides. 1 2 m 2 = (60) = 30 Substitute 60 for m 1. m 2 = m 1The apothem bisects the vertex angle of the isosceles triangle formed by the radii. 1 2 m 3 = 180 – (90 + 30) = 60 The sum of the measures of the angles of a triangle is 180. m 1 = 60, m 2 = 30, and m 3 = 60. Areas of Regular Polygons LESSON 10-3 Additional Examples A portion of a regular hexagon has an apothem and radii drawn. Find the measure of each numbered angle. Quick Check

  2. 1 2 A = apArea of a regular polygon 1 2 A = (37.9)(240) Substitute 37.9 for a and 240 for p. Areas of Regular Polygons LESSON 10-3 Additional Examples Find the area of a regular polygon with twenty 12-in. sides and a 37.9-in. apothem. p = nsFind the perimeter. p = (20)(12) = 240 Substitute 20 for n and 12 for s. A = 4548 Simplify. The area of the polygon is 4548 in.2 Quick Check

  3. Consecutive radii form an isosceles triangle, as shown below, so an apothem bisects the side of the octagon. 1 2 To apply the area formula A = ap, you need to find a and p. Areas of Regular Polygons LESSON 10-3 Additional Examples A library is in the shape of a regular octagon. Each side is 18.0 ft. The radius of the octagon is 23.5 ft. Find the area of the library to the nearest 10 ft2.

  4. Step 1: Find the apothem a. a2 + (9.0)2 = (23.5)2Pythagorean Theorem a2 + 81 = 552.25 Solve for a. a2 = 471.25 a 21.7 Areas of Regular Polygons LESSON 10-3 Additional Examples (continued) Step 2: Find the perimeter p. p = nsFind the perimeter. p = (8)(18.0) = 144 Substitute 8 for n and 18.0 for s, and simplify.

  5. Step 3: Find the area A. A = apArea of a regular polygon A (21.7)(144) Substitute 21.7 for a and 144 for p. A 1562.4 Simplify. 1 2 1 2 Areas of Regular Polygons LESSON 10-3 Additional Examples (continued) To the nearest 10 ft2, the area is 1560 ft2. Quick Check

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