11-4 Areas of Regular Polygons 11-4 Areas of Regular Polygons and Composite Figuresand Composite Figures

You used inscribed and circumscribed figures and found the areas of circles.

• Find areas of regular polygons.

• Find areas of composite figures.

Regular PolygonRegular Polygon

A regular polygon is both

equilateral and equiangular.

Equilateral—all sides are the

same length.

Equiangular—all angles are

the same measure.

Classify each polygon as equilateral, Classify each polygon as equilateral, equiangular and/or regular.equiangular and/or regular.

Equilateral

Regular

Equiangular

Regular Polygon PartsRegular Polygon Parts

The center of a regular polygon is the point of intersection of the perpendicular bisectors of the sides.

center

Regular Polygon PartsRegular Polygon Parts

• The segment from a vertex to the center is the radius.

radius

Regular Polygon PartsRegular Polygon Parts

The perpendicular segment from the center to a side is an apothem (AP uh them).

apothemcenter

All of the radii of a regular polygon are congruent. Any regular polygon can be divided into congruent isosceles triangles.

To find the area, you could then find the area of these triangles.

OR….

Central AngleCentral Angle

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 or XQ

apothem: XN

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. 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.

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.

Step 2 Find the area of one triangle.

b = 3 and

h = 1.7Simplify.

Step 3 Multiply the area of one triangle by the total number of triangles. = 2.55 ft2

Since there are 6 triangles, the area of the table is 2.55 ● 6 or 15.3 ft2.

Answer: 15.3 ft2

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 Regular Polygon TheoremUse the Regular Polygon Theorem

The area of a regular polygon is one-half the product of its perimeter and its apothem.

A = ½ap

p. 808

Find the area of the regular Find the area of the regular pentagon.pentagon.

A pentagon has ___ sides.

The perimeter is ___.

Area = ½ (apothem)(perimeter)

A = ½(6.9)(50)

A = 172.5 in.2

5

5010 in a = 6.9 in

Find the area of a regular octagonFind the area of a regular octagon

An octagon has ___ sides.

The perimeter is ___.

Area = ½ (apothem)(perimeter)

A = ½(18.1)(120)

A = 1086 cm.2

a = 18.1 cm

15cm8

120

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

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

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

Composite FiguresComposite Figures

• A composite figure is a figure that can be separated into regions that are basic figures, such as triangles, rectangles, trapezoids, and circles.

• To find the area of a composite figure, find the area of each basic figure and then use the Area Addition Postulate.

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.

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

Area of composite figure

953.1

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 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.

Answer: The area of the shaded figure is 8500 square feet.

Simplify.

Substitution

Simplify.

Area formulas

A of shaded figure = A of larger rectangle – A of smaller rectangle

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.

Page 811, 8-12, 15-16, 19

No formulas with numbers, no credit.