Areas of Regular Polygons and Composite Figures LESSON 11–4 Areas of Regular Polygons and Composite Figures

Five-Minute Check (over Lesson 11–3) TEKS 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 Lesson Menu

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 5-Minute Check 1

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 5-Minute Check 2

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 5-Minute Check 3

Find the area of the shaded region. Assume that the polygon is regular 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 5-Minute Check 4

Find the area of the shaded region. Assume that the polygon is regular 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 5-Minute Check 5

The area of a circle is 804. 2 square centimeters 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° 5-Minute Check 6

Mathematical Processes G.1(E), G.1(B) Targeted TEKS G.11(A) Apply the formula for the area of regular polygons to solve problems using appropriate units of measure. G.11(B) Determine the area of composite two-dimensional figures comprised of a combination of triangles, parallelograms, trapezoids, kites, regular polygons, or sectors of circles to solve problems using appropriate units of measure. Also addresses G.10(B). Mathematical Processes G.1(E), G.1(B) TEKS

Find areas of regular polygons. You used inscribed and circumscribed figures and found the areas of circles. Find areas of regular polygons. Find areas of composite figures. Then/Now

center of a regular polygon radius of a regular polygon apothem central angle of a regular polygon composite figure Vocabulary

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 Example 1

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: mRXQ = 72° Example 1

In the figure, hexagon ABCDEF is inscribed in Find the measure of a central angle. A. mDGH = 45° B. mDGC = 60° C. mCGD = 72° D. mGHD = 90° Example 1

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. Example 2

Step 2 Find the area of one triangle. Area of a Regular Polygon Step 2 Find the area of one triangle. Area of a triangle b = 3 and h = 1.7 Simplify. = 2.55 ft2 Step 3 Multiply the area of one triangle by the total number of triangles. Example 2

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 Example 2

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 Example 2

Concept

A. Find the area of the regular hexagon. Round to the nearest tenth. 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 Example 3A

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 mSPR = 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 Example 3A

Step 3 Use the apothem and side length to find the area. 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 Use a calculator. Answer: about 65.0 m2 Example 3A

B. Find the area of the regular pentagon. Round to the nearest tenth. 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 Example 3B

Use the Formula for the Area of a Regular Polygon Step 2 Apothem CD is the height of isosceles ΔBCA. It bisects BCA, so mBCD = 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°. Example 3B

Step 3 Area of a regular polygon 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 Example 3B

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 Example 3

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 Example 3

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? 1556.2 ft2 1193.1 ft2 953.1 ft2 852.5 ft2 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. Example 4

Area of composite figure 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. The answer is C. Example 4

Find the area of the figure in square feet 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 Example 4

Find the area of the shaded figure. 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. Example 5

= area of larger rectangle – area of smaller rectangle 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. Example 5

INTERIOR DESIGN Cara wants to wallpaper one wall of her family room 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 Example 5

Areas of Regular Polygons and Composite Figures LESSON 11–4 Areas of Regular Polygons and Composite Figures