Finding Perimeters and Areas of Regular Polygons
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1 Finding Perimeters and Areas of Regular Polygons
2 Center of a Regular Polygon - A point within the polygon that is equidistant from all vertices. Central Angle of a Regular Polygon - The angle whose vertex is the center of a regular polygon and whose sides pass through consecutive vertices.
3 Apothem - The perpendicular distance from the center of a regular polygon to a side. You can use the formula P = ns to find the perimeter of a regular polygon. In the formula, P represents the perimeter, n represents the number of sides, and s represents the side length.
4 a. Find the perimeter of the polygon. SOLUTION P = ns Formula for perimeter of a regular polygon P = (6)(3) Substitute. P = 18 ft
5 b. Find the perimeter of the polygon. SOLUTION P = ns Formula for perimeter of a regular polygon P = Substitute. P = 27 cm
6 You can find the area, A, of a regular polygon using only the apothem and perimeter. Consider an n-sided regular polygon with a side length of s. Divide the polygon into n triangles so the vertices of each triangle are the center of the polygon and the endpoints of a side as shown. By definition, the height of each triangle is the apothem, a. The base of each triangle has a length of s. So, the area of each 1 triangle is as. The total area of the polygon is n times the area of 2 one triangle, or A = 1 2 nas The formula for the perimeter of a regular polygon is P = ns. By substitution, the area of a regular polygon is A = 1 2 ap
7 Area Formula for Regular Polygons - The area, A, of a regular polygon is half the apothem length a and the perimeter P of the regular polygon. A = 1 2 ap
8 Find the area of a regular octagon with an apothem about 18 inches. SOLUTION P = ns Formula for perimeter of a regular polygon P = (8)(15) Substitute. P = 120 Simplify. A = 1 ap Area formula for regular polygons 2 A = Substitute. 2 A = 1080 Simplify. The area is 1080 square inches.
9 Use the apothem and perimeter to find the area of this regular hexagon. SOLUTION First, find the apothem length of the regular hexagon. Draw an isosceles triangle whose vertices are the center of the hexagon and the endpoints of a side. The triangle contains a central angle whose measure is 60. The apothem bisects the central angle and the side, forming a triangle. The shorter leg of the triangle is 5 centimeters long. Therefore, the apothem measures 5 3 centimeters.
10 Use the apothem and perimeter to find the area of this regular hexagon. SOLUTION Next, find the perimeter of the hexagon. P = ns Formula for perimeter of a regular polygon P = (6)(10) Substitute. P = 60 Simplify. Finally, find the area of the hexagon. A = 1 ap Area formula for regular polygons 2 A = Substitute. 2 A = Simplify. The area is centimeters squared.
11 Find the area of an equilateral triangle with 18-inch sides. SOLUTION Use your knowledge of triangles to find the apothem, a, and then use your result to find the area. a 9 = 1 3 a = 9 3 a = 3 3 A = 1 2 ap A = A = 81 3
12 A plot of land is in the shape of a regular octagon with 10-mile side lengths and apothem of about 12 miles. The plot needs be divided into eight equal parcels of land. What will the area of land be in each parcel? SOLUTION First, find the area of the plot of land. Because the plot is in the shape of an octagon where each side is 10 miles long, its perimeter is 80 miles. A = 1 2 ap Area formula for regular polygons A = Substitute. 2 A = 480 Simplify. The area of the octagonal plot is 480 mi 2. Next, divide the total area by 8 to find the area in each equal parcel = 60 Each of the 8 parcels has an area of about 60 mi 2.
13 a. Find the perimeter of this octagon. b. Use the area formula for regular polygons to find the area of this pentagon.
14 c. Find the area of this hexagon. d. Find the area of this equilateral triangle.
15 e. The shape of a playground is a regular hexagon where each side length is 78 feet long. The playground is to be resurfaced with a nonslip rubber material. What is the total area that must be surfaced?
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CHAPTER 10 Perimeter and Area Chapter Outline 10.1 TRIANGLES AND PARALLELOGRAMS 10.2 TRAPEZOIDS, RHOMBI, AND KITES 10.3 AREAS OF SIMILAR POLYGONS 10.4 CIRCUMFERENCE AND ARC LENGTH 10.5 AREAS OF CIRCLES