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The area is about 695.29 square units.
A = 2(1 + sqrt2)S2 where S is a side length
P = 8*S
S = 12
A = (2 + 2 sqrt2) 144
A = 288 + 288 sqrt2
A = 288 + 407.29
A = 695.29
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Derivation
The area of a regular polygon is:
Area = (1/2)(p)(a), where p is the perimeter and a the apothem.
Since we know the perimeter, and also the length of each side of our octagon which is 12 (96/8), we need to find the length of the apothem a.
Let's take a look at one of the right triangle which is formed when we draw the radius from the center to one of the vertices, and the apothem a from the center perpendicular to the side s.
In this right triangle we have:
side s = 6 (12/2)
angle opposite to apothem = 67.5 degrees (135/2)
So that,
tan 67.5 degrees = a/s
a = (tan 67.5 degrees)(6)
a = 14.5 (approximately)
Thus,
Area = (1/2)(p)(a)
Area = (1/2)(96)(14.5)
Area = 696
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∙ 11y ago
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What are the dimensions of a rectangle that has a perimeter of 56 units and an area of 96 square units
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