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Regular pentagon DEFGH has a center C , apothem BC , and central angle  DCE .

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Regular pentagon DEFGH has a center C , apothem BC , and central angle  DCE .

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  1. The center of a regular polygonis equidistant from the vertices. The apothemis the distance from the center to a side. A central angle of a regular polygonhas its vertex at the center, and its sides pass through consecutive vertices. Each central angle measure of a regular n-gon is

  2. Regular pentagon DEFGH has a center C, apothem BC, and central angle DCE.

  3. area of each triangle: total area of the polygon: To find the area of a regular n-gon with side length s and apothem a, divide it into n congruent isosceles triangles. The perimeter is P = ns.

  4. Step 1 Draw the heptagon. Draw an isosceles triangle with its vertex at the center of the heptagon. The central angle is . Example 3A: Finding the Area of a Regular Polygon Find the area of regular heptagon with side length 2 ft to the nearest tenth. Draw a segment that bisects the central angle and the side of the polygon to form a right triangle.

  5. The tangent of an angle is . opp. leg adj. leg Example 3A Continued Step 2 Use the tangent ratio to find the apothem. Solve for a.

  6. Example 3A 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. Simplify. Round to the nearest tenth. A 14.5 ft2

  7. Remember! The tangent of an angle in a right triangle is the ratio of the opposite leg length to the adjacent leg length. See page 525.

  8. Step 1 Draw the dodecagon. Draw an isosceles triangle with its vertex at the center of the dodecagon. The central angle is . Example 3B: Finding the Area of a Regular Polygon Find the area of a regular dodecagon with side length 5 cm to the nearest tenth. Draw a segment that bisects the central angle and the side of the polygon to form a right triangle.

  9. The tangent of an angle is . opp. leg adj. leg Example 3B Continued Step 2 Use the tangent ratio to find the apothem. Solve for a.

  10. Example 3B 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. Simplify. Round to the nearest tenth. A 279.9 cm2

  11. Step 1 Draw the octagon. Draw an isosceles triangle with its vertex at the center of the octagon. The central angle is . Check It Out! Example 3 Find the area of a regular octagon with a side length of 4 cm. Draw a segment that bisects the central angle and the side of the polygon to form a right triangle.

  12. The tangent of an angle is . opp. leg adj. leg Check It Out! Example 3 Continued Step 2 Use the tangent ratio to find the apothem Solve for a.

  13. Check It Out! Example 3 Continued 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. Simplify. Round to the nearest tenth. A ≈ 77.3 cm2

  14. Lesson Quiz: Part II Find each measurement. 3. Speakers come in diameters of 4 in., 9 in., and 16 in. Find the area of each speaker to the nearest tenth. A1≈ 12.6 in2 ; A2≈ 63.6 in2 ; A3≈ 201.1 in2 Find the area of each regular polygon to the nearest tenth. 4. a regular nonagon with side length 8 cm A ≈ 395.6 cm2 5. a regular octagon with side length 9 ft A ≈ 391.1 ft2

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