EXAMPLE 2 Find the area of a regular polygon

DECORATING

You are decorating the top of a table by covering it with small ceramic tiles. The table top is a regular octagon with 15 inch sides and a radius of about 19.6 inches. What is the area you are covering?

SOLUTION

STEP 1 Find the perimeter P of the table top. An octagon has 8 sides, so P = 8(15) = 120 inches.

EXAMPLE 2

STEP 2

So, QS = (QP) = (15) = 7.5 inches.12

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To find RS, use the Pythagorean Theorem for ∆ RQS.

a = RS ≈ √19.62 – 7.52 = 327.91 ≈ 18.108 √

Find the apothem a. The apothem is height RS of ∆PQR. Because ∆PQR is isosceles, altitude RS bisects QP .

Find the area of a regular polygon

EXAMPLE 2

STEP 3 Find the area A of the table top.12A = aP Formula for area of regular polygon

≈ (18.108)(120)12 Substitute.

≈ 1086.5 Simplify.

Find the area of a regular polygon

So, the area you are covering with tiles is about 1086.5 square inches.

ANSWER

EXAMPLE 3 Find the perimeter and area of a regular polygon

A regular nonagon is inscribed in a circle with radius 4 units. Find the perimeter and area of the nonagon.

SOLUTION360°

The measure of central JLK is , or 40°. Apothem LM bisects the central angle, so m KLM is 20°. To find the lengths of the legs, use trigonometric ratios for right ∆ KLM.

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EXAMPLE 3

sin 20° = MKLK

sin 20° = MK4

4 sin 20° = MK

cos 20° = LMLK

cos 20° = LM4

4 cos 20° = LM

The regular nonagon has side length s = 2MK = 2(4 sin 20°) = 8(sin 20°) and apothem a = LM = 4(cos 20°).

Find the perimeter and area of a regular polygon

EXAMPLE 3 Find the perimeter and area of a regular polygon

So, the perimeter is P = 9s = 9(8 sin 20°) = 72 sin 20° ≈ 24.6 units,and the area is A = aP = (4 cos 20°)(72 sin 20°) ≈ 46.3 square units.

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ANSWER

GUIDED PRACTICE for Examples 2 and 3

3.

SOLUTION

The measure of the central angle is = or 72°. Apothem a bisects the central angle, so angle is 36°. To find the lengths of the legs, use trigonometric ratios for right angle.

3605

Find the perimeter and the area of the regular polygon.

GUIDED PRACTICE for Examples 2 and 3

sin 36° = bhyp

sin 36° = b8

So, the perimeter is P = 5s = 5(16 sin 36°)

8 sin 36° = b

The regular pentagon has side length = 2b = 2(8 sin 36°) = 16 sin 36° 20°

= 80 sin 36°

≈ 46.6 units,

and the area is A = aP = 6.5 46.6

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≈ 151.5 units2.

GUIDED PRACTICE for Examples 2 and 3

4.

SOLUTION

The regular nonagon has side length = 7.

So, the perimeter is P = 10 · s = 10 · 7 = 70 units

Find the perimeter and the area of the regular polygon.

GUIDED PRACTICE for Examples 2 and 3

and the area is A = aP = 10.8 70

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≈ 377 units2.

The measure of central is = or 36°. Apothem a bisects the central angle, so angle is 18°. To find the lengths of the legs, use trigonometric ratios for right angle.

36010

tan 18° =oppadj

tan 18° = 3.5a

a =

3.5tan 18° ≈10.8

GUIDED PRACTICE for Examples 2 and 3

5.

SOLUTION

The measure of central angle is = 120°. Apothem a bisects the central angle, so is 60°. To find the lengths of the legs, use the trigonometric ratios.

360°3

GUIDED PRACTICE for Examples 2 and 3

cos 60° = ax

x = 10

sin 60° = b10

The regular polygon has side length s = 2 = 2 (10 sin 60°) = 20 sin 60° and apothem a = 5.

x cos 60° = 5

x 0.5 = 5

b10 sin 60° =

GUIDED PRACTICE for Examples 2 and 3

= 30 3 units

= 60 sin 60°

and the area is A = aP12

= × 5 30 312

= 129.9 units2

So, the perimeter is P = 3 s = 3(20 sin 60°)

GUIDED PRACTICE for Examples 2 and 3

6. Which of Exercises 3–5 above can be solved using special right triangles?

Exercise 5 can be solved using special right triangles. The triangle is a 30-60-90 Right Triangle

ANSWER