Question

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

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

Step-by-step 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°.

__

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.

__

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.

__

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.

__

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.