#### Transcript Area of a Regular Polygon

Five-Minute Check (over Lesson 11–3) CCSS Then/Now New Vocabulary Example 1: Identify Segments and Angles in Regular Polygons Example 2: Real-World Example: Area of a Regular Polygon Key Concept: Area of a Regular Polygon Example 3: Use the Formula for the Area of a Regular Polygon Example 4: Find the Area of a Composite Figure by Adding Example 5: Find the Area of a Composite Figure by Subtracting Over Lesson 11–3 Find the area of the circle. Round to the nearest tenth. A. 37.7 ft2 B. 75.4 ft2 C. 223.6 ft2 D. 452.4 ft2 Over Lesson 11–3 Find the area of the sector. Round to the nearest tenth. A. 25.1 m2 B. 28.3 m2 C. 33.4 m2 D. 50.2 m2 Over Lesson 11–3 Find the area of the sector. Round to the nearest tenth. A. 506.8 in2 B. 570.2 in2 C. 760.3 in3 D. 1520.5 in2 Over Lesson 11–3 Find the area of the shaded region. Assume that the polygon is regular. Round to the nearest tenth. A. 36.4 units2 B. 39.1 units2 C. 47.3 units2 D. 51.4 units2 Over Lesson 11–3 Find the area of the shaded region. Assume that the polygon is regular. Round to the nearest tenth. A. 82.5 units2 B. 87.3 units2 C. 92.5 units2 D. 106.7 units2 Over Lesson 11–3 The area of a circle is 804.2 square centimeters. The area of a sector of the circle is 268.1 square centimeters. What is the measure of the central angle that defines the sector? A. 110° B. 120° C. 135° D. 150° Content Standards G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Mathematical Practices 1 Make sense of problems and persevere in solving them. 6 Attend to precision. You used inscribed and circumscribed figures and found the areas of circles. • Find areas of regular polygons. • Find areas of composite figures. • center of a regular polygon • radius of a regular polygon • apothem • central angle of a regular polygon • composite figure Identify Segments and Angles in Regular Polygons In the figure, pentagon PQRST is inscribed in Identify the center, a radius, an apothem, and a central angle of the polygon. Then find the measure of a central angle. center: point X radius: XR or XQ apothem: XN central angle: RXQ Identify Segments and Angles in Regular Polygons A pentagon is a regular polygon with 5 sides. Thus, the measure of each central angle of pentagon PQRST is or 72. Answer: mRXQ = 72° In the figure, hexagon ABCDEF is inscribed in Find the measure of a central angle. A. mDGH = 45° B. mDGC = 60° C. mCGD = 72° D. mGHD = 90° Area of a Regular Polygon FURNITURE The top of the table shown is a regular hexagon with a side length of 3 feet and an apothem of 1.7 feet. What is the area of the tabletop to the nearest tenth? Step 1 Since the polygon has 6 sides, the polygon can be divided into 6 congruent isosceles triangles, each with a base of 3 ft and a height of 1.7 ft. Area of a Regular Polygon Step 2 Find the area of one triangle. Area of a triangle b = 3 and h = 1.7 = 2.55 ft2 Simplify. Step 3 Multiply the area of one triangle by the total number of triangles. Area of a Regular Polygon Since there are 6 triangles, the area of the table is 2.55 ● 6 or 15.3 ft2. Answer: 15.3 ft2 UMBRELLA The top of an umbrella shown is a regular hexagon with a side length of 2 feet and an apothem of 1.5 feet. What is the area of the entire umbrella to the nearest tenth? A. 6 ft2 B. 7 ft2 C. 8 ft2 D. 9 ft2 Use the Formula for the Area of a Regular Polygon A. Find the area of the regular hexagon. Round to the nearest tenth. Step 1 Find the measure of a central angle. A regular hexagon has 6 congruent central angles, so Use the Formula for the Area of a Regular Polygon Step 2 Find the apothem. Apothem PS is the height of isosceles ΔQPR. It bisects QPR, so mSPR = 30. It also bisects QR, so SR = 2.5 meters. ΔPSR is a 30°-60°-90° triangle with a shorter leg that measures 2.5 meters, so Use the Formula for the Area of a Regular Polygon Step 3 Use the apothem and side length to find the area. Area of a regular polygon ≈ 65.0 m2 Answer: about 65.0 m2 Use a calculator. Use the Formula for the Area of a Regular Polygon B. Find the area of the regular pentagon. Round to the nearest tenth. Step 1 A regular pentagon has 5 congruent central angles, so Use the Formula for the Area of a Regular Polygon Step 2 Apothem CD is the height of isosceles ΔBCA. It bisects BCA, so mBCD = 36. Use trigonometric ratios to find the side length and apothem of the polygon. AB = 2DB or 2(9 sin 36°). So, the pentagon’s perimeter is 5 ● 2(9 sin 36°). The length of the apothem CD is 9 cos 36°. Use the Formula for the Area of a Regular Polygon Step 3 Area of a regular polygon a = 9 cos 36° and P = 10(9 sin 36°) Use a calculator. Answer: 192.6 cm2 A. Find the area of the regular hexagon. Round to the nearest tenth. A. 73.1 m2 B. 96.5 m2 C. 126.8 m2 D. 146.1 m2 B. Find the area of the regular pentagon. Round to the nearest tenth. A. 116.5 m2 B. 124.5 m2 C. 138.9 m2 D. 143.1 m2 Find the Area of a Composite Figure by Adding POOL The dimensions of an irregularly shaped pool are shown. What is the area of the surface of the pool? The figure can be separated into a rectangle with dimensions 16 feet by 32 feet, a triangle with a base of 32 feet and a height of 15 feet, and two semicircles with radii of 8 feet. Find the Area of a Composite Figure by Adding Area of composite figure 953.1 Answer: The area of the composite figure is 953.1 square feet to the nearest tenth. Find the area of the figure in square feet. Round to the nearest tenth if necessary. A. 478.5 ft2 B. 311.2 ft2 C. 351.2 ft2 D. 438.5 ft2 Find the Area of a Composite Figure by Subtracting Find the area of the shaded figure. To find the area of the figure, subtract the area of the smaller rectangle from the area of the larger rectangle. The length of the larger rectangle is 25 + 100 + 25 or 150 feet. The width of the larger rectangle is 25 + 20 + 25 or 70 feet. Find the Area of a Composite Figure by Subtracting area of shaded figure = area of larger rectangle – area of smaller rectangle Area formulas Substitution Simplify. Simplify. Answer: The area of the shaded figure is 8500 square feet. INTERIOR DESIGN Cara wants to wallpaper one wall of her family room. She has a fireplace in the center of the wall. Find the area of the wall around the fireplace. A. 168 ft2 B. 156 ft2 C. 204 ft2 D. 180 ft2