example 2
DESCRIPTION
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?. Find the area of a regular polygon. EXAMPLE 2. DECORATING. SOLUTION. STEP 1. - PowerPoint PPT PresentationTRANSCRIPT
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
12
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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°)