Over Lesson 11–3 A.A B.B C.C D.D 5-Minute Check ft 2 Find the area of the circle. Round to the nearest tenth.

Over Lesson 11–3 A.A B.B C.C D.D 5-Minute Check m 2 Find the area of the sector. Round to the nearest tenth.

Over Lesson 11–3 A.A B.B C.C D.D 5-Minute Check in 2 Find the area of the sector. Round to the nearest tenth.

Over Lesson 11–3 A.A B.B C.C D.D 5-Minute Check units 2 Find the area of the shaded region. Assume that the polygon is regular. Round to the nearest tenth.

Over Lesson 11–3 A.A B.B C.C D.D 5-Minute Check units 2 Find the area of the shaded region. Assume that the polygon is regular. Round to the nearest tenth.

Over Lesson 11–3 A.A B.B C.C D.D 5-Minute Check 6 120° The area of a circle is square centimeters. The area of a sector of the circle is square centimeters. What is the measure of the central angle that defines the sector?

Then/Now Find areas of regular polygons. Find areas of composite figures.

Vocabulary center of a regular polygon--the point that is equidistant from each vertex radius of a regular polygon-- the distance from the center to any vertex. apothem--A segment from the center of a regular polygon perpendicular to a side of the polygon.

central angle of a regular polygon--the angle made at the center of the polygon by any two adjacent vertices of the polygon composite figure-- a figure is made from two or more geometric figures

Example 1 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 central angle:  RXQ radius: XR and XQ apothem: XN

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°

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

Example 2 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 1Since 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 Area of a Regular Polygon Step 2Find the area of one triangle. Area of a triangle b = 3 and h = 1.7 Simplify. Step 3Multiply the area of one triangle by the total number of triangles. = 2.55 ft 2

Example 2 Area of a Regular Polygon Since there are 6 triangles, the area of the table is 2.55 ● 6 or 15.3 ft 2. Answer: 15.3 ft 2

A.A B.B C.C D.D Example 2 9 ft 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?

Concept

Example 3A Use the Formula for the Area of a Regular Polygon A. Find the area of the regular hexagon. Round to the nearest tenth. Step 1Find 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 2Find 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 Use the Formula for the Area of a Regular Polygon Step 3Use the apothem and side length to find the area. Area of a regular polygon ≈ 65.0 m 2 Use a calculator. Answer: about 65.0 m 2

Example 3B Use the Formula for the Area of a Regular Polygon B. Find the area of the regular pentagon. Round to the nearest tenth. Step 1A regular pentagon has 5 congruent central angles, so

Example 3B Use the Formula for the Area of a Regular Polygon Step 2Apothem 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 Use the Formula for the Area of a Regular Polygon Step 3Area of a regular polygon a = 9 cos 36° and P = 10(9 sin 36°) Use a calculator. Answer: cm 2

A.A B.B C.C D.D Example m 2 A. Find the area of the regular hexagon. Round to the nearest tenth.

A.A B.B C.C D.D Example m 2 B. Find the area of the regular pentagon. Round to the nearest tenth.

Example 4 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.

Example 4 Find the Area of a Composite Figure by Adding Answer: The area of the composite figure is square feet to the nearest tenth. Area of composite figure = 953.1

A.A B.B C.C D.D Example ft 2 Find the area of the figure in square feet. Round to the nearest tenth if necessary.

Example 5 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 or 150 feet. The width of the larger rectangle is or 70 feet.

Example 5 Find the Area of a Composite Figure by Subtracting Answer:The area of the shaded figure is 8500 square feet. Simplify. Substitution Simplify. Area formulas area of shaded figure = area of larger rectangle – area of smaller rectangle

A.A B.B C.C D.D Example ft 2 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.