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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°.
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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.
