Five-Minute Check (over Lesson 11–3)

NGSSS

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

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. 37.7 ft2

B. 75.4 ft2

C. 223.6 ft2

D. 452.4 ft2

Find the area of the circle.Round to the nearest tenth.

Over Lesson 11–3

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. 25.1 m2

B. 28.3 m2

C. 33.4 m2

D. 50.2 m2

Find the area of the sector.Round to the nearest tenth.

Over Lesson 11–3

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. 506.8 in2

B. 570.2 in2

C. 760.3 in3

D. 1520.5 in2

Find the area of the sector.Round to the nearest tenth.

Over Lesson 11–3

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. 36.4 units2

B. 39.1 units2

C. 47.3 units2

D. 51.4 units2

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 A B C D

0% 0%0%0%

A. 82.5 units2

B. 87.3 units2

C. 92.5 units2

D. 106.7 units2

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 A B C D

0% 0%0%0%

A. 110°

B. 120°

C. 135°

D. 150°

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?

MA.912.G.2.5 Explain the derivation and apply formulas for perimeter and area of polygons.

MA.912.G.2.6 Use coordinate geometry to prove properties of congruent, regular and similar polygons, and to perform transformations in the plane.

Also addresses MA.912.G.2.7.

You used inscribed and circumscribed figures and found the areas of circles. (Lessons 10–1and 11–3)

• 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

central angle: RXQ

radius: XR and XQ

apothem: XN

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°

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. mDGH = 45°

B. mDGC = 60°

C. mCGD = 72°

D. mGHD = 90°

In the figure, hexagon ABCDEF is inscribed in Find the measure of a central angle.

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 polygoncan be divided into 6 congruent isoscelestriangles, each with a base of 3 ft and aheight 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

Simplify.

Step 3 Multiply the area of one triangle by the totalnumber of triangles.

= 2.55 ft2

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

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. 6 ft2

B. 7 ft2

C. 8 ft2

D. 9 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?

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 isoscelesΔQPR. It bisects QPR, so mSPR = 30. It also bisects QR, so SR = 2.5 meters.

ΔPSR is a 30°-60°-90° triangle with ashorter 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 Use a calculator.

Answer: about 65.0 m2

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 isoscelesΔBCA. It bisects BCA, so mBCD = 36.Use trigonometric ratios to find the sidelength 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. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. 73.1 m2

B. 96.5 m2

C. 126.8 m2

D. 146.1 m2

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

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. 116.5 m2

B. 124.5 m2

C. 138.9 m2

D. 143.1 m2

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

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

Answer: The area of the composite figure is 953.1 square feet to the nearest tenth.

Area of composite figure

= 953.1

A. A

B. B

C. C

D. D A B C D

0% 0%0%0%

A. 478.5 ft2

B. 311.2 ft2

C. 351.2 ft2

D. 438.5 ft2

Find the area of the figure in square feet. Round to the nearest tenth if necessary.

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

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 A B C D

0% 0%0%0%

A. 168 ft2

B. 156 ft2

C. 204 ft2

D. 180 ft2

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.