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3 11-3 Objectives You will learn to: You will learn to find the area of a regular polygon.
4 Vocabulary Center of a regular polygon Apothem Central angle of a regular polygon
5 Definitions The center of a regular polygon is equidistant from the vertices. The apothem is the distance from the center to a side. A central angle of a regular polygon has its vertex at the center, and its sides pass through consecutive vertices. Each central angle measure of a regular n-gon is
6 Finding Area Regular pentagon DEFGH has a center C, apothem BC, and central angle DCE. To find the area of a regular n-gon with side length s and apothem a, divide it into n congruent isosceles triangles. Area of each triangle: Total area of the polygon: The perimeter is P = ns.
7 Remember! The tan of an angle in a right triangle is the ratio of the opposite leg length to the adjacent leg length.
8 Find the area of a regular pentagon with a perimeter of 90 meters.
9 Apothem: The central angles of a regular pentagon are all congruent. Therefore, the measure of each angle is or 72. is an apothem of pentagon ABCDE. It bisects and is a perpendicular bisector of. So, or 36. Since the perimeter is 90 meters, each side is 18 meters and meters.
10 Write a trigonometric ratio to find the length of. Multiply each side by GF. Divide each side by tan. Use a calculator.
11 Area: Area of a regular polygon Simplify. Answer: The area of the pentagon is about 558 square meters.
12 Find the area of a regular pentagon with a perimeter of 120 inches. Answer: about
13 Find the area of regular heptagon with side length 2 ft to the nearest tenth. Step 1 Draw the heptagon. Draw an isosceles triangle with its vertex at the center of the heptagon. The central angle is. Draw a segment that bisects the central angle and the side of the polygon to form a right triangle. Step 2 Use the tangent ratio to find the apothem. The tangent of an angle is opp. leg. adj. leg Solve for a.
14 Continued Step 3 Use the apothem and the given side length to find the area. Area of a regular polygon The perimeter is 2(7) = 14ft. A 14.5 ft 2 Simplify. Round to the nearest tenth.
15 Find the area of a regular dodecagon with side length 5 cm to the nearest tenth. Step 1 Draw the dodecagon. Draw an isosceles triangle with its vertex at the center of the dodecagon. The central angle is. Draw a segment that bisects the central angle and the side of the polygon to form a right triangle. Step 2 Use the tangent ratio to find the apothem. opp. leg The tangent of an angle is. adj. leg Solve for a.
16 Continued Step 3 Use the apothem and the given side length to find the area. Area of a regular polygon The perimeter is 5(12) = 60 ft. A cm 2 Simplify. Round to the nearest tenth.
17 Find the area of a regular octagon with a side length of 4 cm. Your Turn Step 1 Draw the octagon. Draw an isosceles triangle with its vertex at the center of the octagon. The central angle is. Draw a segment that bisects the central angle and the side of the polygon to form a right triangle. Step 2 Use the tangent ratio to find the apothem The tangent of an angle is. opp. leg adj. leg Solve for a.
18 Your Turn Step 3 Use the apothem and the given side length to find the area. Area of a regular polygon The perimeter is 4(8) = 32cm. A 77.3 cm 2 Simplify. Round to the nearest tenth.
19 What did you learn today? How to: You will learn to find the area of a regular polygon.
20 Assignment: Page , 26, 27
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