What is the area of a regular octagon with a perimeter of 96?

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