Question

● For the polygon you are assigned, assume the radius is 1 unit . Round all answers to the nearest hundredth , if necessary.

1) Calculate the measure of each interior angle of your polygon. Be sure to show your work!

2) Use your knowledge of trigonometry to find the length of one side of your polygon. Be sure to show your work!

3) Use your knowledge of trigonometry to find the length of the apothem (the segment from the center to the midpoint of one side). Be sure to show your work!

4) Calculate the perimeter of your polygon. Be sure to show your work!

5) Calculate the area of your polygon by splitting it into congruent isosceles triangles. Be sure to show your work!

Answer 1

9514 1404 393

Answer:

1) 135°

2) 0.77 units

3) 0.92 units

4) 6.12 units

5) 2.83 square units

Explanation:

1) The exterior angle at any vertex is 360°/n, where n is the number of sides. For the octagon, the exterior angle is 360°/8 = 45°. The interior angle will be the supplement of this: 180° -45° = 135°.

The measure of each interior angle is 135°.

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2) Each sector of the octagon is an isosceles triangle with a central angle of 45° (also 360°/8). Since the radius is 1 unit, the length of one side is twice the sine of half the central angle.

s = 2·sin(45°/2) ≈ 0.765367

The length of one side is about 0.77 units.

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3) Similarly, the apothem is the cosine of half the central angle:

a = cos(45°/2) ≈ 0.923880

The apothem is about 0.92 units.

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4) For an octagon, the perimeter is 8 times the length of one side.

P = 8s = 8(0.765367) ≈ 6.12293

The perimeter is about 6.12 units.

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5) The area of one sector (isosceles triangle) is given by the formula ...

A = (1/2)bh

A = 1/2sa ≈ 1/2(0.765367)(0.923880) ≈ 0.353553

Then the area of the octagon is 8 times this:

A = 8(sector area) = 8(0.353553) ≈ 2.82843

The octagon area is about 2.83 square units.