#### POLYGON AND CIRCLE EXERCISES

QUESTIONS

1/ Find the length of the arc AB and the area of the sector AOB. (click for explanation)

Write the formula for arc length and sector area in each solution.

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2) a) The area of a 40° sector of a circle is 15.6 dm2. Find the area of whole circle.

b) Find the area of a sector of a circle formed by an angle of 72° if the area of the
whole circle is 255 cm 2.

c) Circle center O has a diameter of 12 cm. Find:

 i) the circumference ii) the area iii) Find the length of an arc of this circle for a central angle of 72° . iv) Find the area of the sector. (click for solution)

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3) a) A regular hexagon has an area of 24 cm 2 and the apothem measures 3 cm.

 i) How long is each side? ii) What is the perimeter of this hexagon?

b) Find the area of a regular nonagon if the length of a side is 5.3 dm and the length of a line connecting the center to a vertex is 7 dm.

c) Find the area of the following regular polygons:

 i) pentagon: side = 8.2 m, apothem = 5 m. ii) hexagon: side = 9.4 dm, apothem = 3.2 dm.

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4) Explain how you would construct a regular hexagon with sides of 7 cm.

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5) Find the measure of each interior and exterior angle for:

 a) A regular decagon. b) A regular dodecagon (12 sides) c) A regular septagon.

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SOLUTIONS

A central angle of 360° makes an arc of 2 r units,
so
a central angle of x° makes an arc of

units.

Since a central angle of 360° makes the area = r² units²,
the area of a sector with central angle equal to x° is

units².

 a) arc AB = 12.28 cm. sector area = 39.3 cm² b) arc AB = 10.38 cm. sector area = 88.27 cm² c) arc AB = 52.33 dm.sector area = 654.5 cm²

2)

a) Since 40° is one ninth of a full circle, the total area = 9 ( 15.6) = 140.4 cm².

b) Since 72° is one fifth of a full circle, the sector area = 1/5 ( 255) = 51 cm².

c) diameter = 12 cm. so radius = 6 cm.

 i) the circumference = d = 12 = 37.68 cm. ii) the area = r² = 36 = 113.04 cm². iii) arc for a central angle of 72° = 7.54 cm. iv) Find the area of the sector = 22.61 cm².

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3) a) A regular hexagon with area = 24 cm 2, apothem = 3 cm.

 i) side = 2.67 cm ii) perimeter = 16 cm.

b) area of nonagon if side = 5.3 dm and radius = 7 dm.

Use Pythagoras to solve for the apothem = 6.48 dm. so area = 154.55 dm².

c) Find the area of the following regular polygons:

 i) pentagon: side = 8.2 m, apothem = 5 m. Area = 1025 m². ii) hexagon: side = 9.4 dm, apothem = 3.2 dm.Area = 90.24 dm².

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4) A hexagon has 6 sides. We draw a line segment and mark off AB = 7 cm.
Since each interior angle of a regular hexagon = 120°, we construct a 120° angle
at both A and B. Then we measure 7 cm to C on the right and F on the left. We contiinue
the process until we complete the hexagon.

5) Find the measure of each interior and exterior angle for:

 a) A regular decagon.interior = 144° exterior = 36° b) A regular dodecagon (12 sides)interior = 150° exterior = 30° c) A regular septagon.interior = 128.57° exterior = 51.43°

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