Bellwork Find the values of x and y for each 1. 2. 3. 10 4 y y y 5 30° 1. 2. 3. 10 y 4 y 5 y 30° 30° 45° x x x 10-2

10.2 Area of Regular Polygons Radius: The distance from the center to a vertex Apothem: The perpendicular distance from the center to a side 10-2

Area of Regular Polygons A regular hexagon has an apothem and radii drawn. Find the measure of each numbered angle. 1 2 m 1 = = 60 Divide 360 by the number of sides. 360 6 3 m 2 = m 1 The apothem bisects the vertex angle of the isosceles triangle formed by the radii. 1 2 m 2 = (60) = 30 Substitute 60 for m 1. 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. 10-2

Area of Regular Polygons A regular octagon has an apothem and radii drawn. Find the measure of each numbered angle. 3 1 2 10-2

Area of Regular Polygons A square has an apothem and radii drawn. Find the measure of each numbered angle. 2 1 3 10-2

Area of Regular Polygons Area of a regular Polygon The area of a regular polygon is half the product of the apothem and the perimeter Area = ½ap 10-2

Area of Regular Polygons Find the area of a regular polygon with twenty 12-in. sides and a 37.9-in. apothem. A = ap Area of a regular polygon 1 2 A = (37.9)(p) Substitute 37.9 for a. 1 2 Find the perimeter p = ns p = (20)(12) = 240 Substitute 20 for n and 12 for s. A = (37.9)(240) Substitute 240 for p. 1 2 A = 4548 Simplify. The area of the polygon is 4548 in.2 10-2

Area of Regular Polygons Find the area of an equilateral triangle with apothem 8 cm. Leave your answer in simplest radical form. 8 10-2

Area of Regular Polygons Find the area of an regular hexagon with radius 10m. Leave your answer in simplest radical form. 10-2

Area of Regular Polygons 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. To apply the area formula A = ap, you need to find a and p. 1 2 Consecutive radii form an isosceles triangle, so an apothem bisects the side of the octagon. 10-2

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

Area of Regular Polygons (continued) Step 3: Find the area A. A = ap Area of a regular polygon A (21.7)(144) Substitute 21.7 for a and 144 for p. A 1562.4 Simplify. 1 2 To the nearest 10 ft2, the area is 1560 ft2. 10-2

Composite Figures A composite figure is a figure comprised of simple shapes (rectangle, circle, parallelogram, etc.) 10-3

Composite Figures To find the area of a composite figure divide the figure into non-overlapping shapes of polygons that we have formulas for their areas. 7 10 4 3 8 6 10-3

Composite Figures 10 4 3 8 10-3

Composite Figures A company receives an order for 65 pieces of fabric in the given shape. Each piece is to be dyed red. To dye 6 in2 of fabric, 2 oz of dye is needed. How much dye is needed for the entire order? 10-3

Composite Figures 10-3

Area of Regular Polygons HOMEWORK 10.2(691): 14,26,29 10.3(697):15,17,19,20