Answer:
173.82 cm²
Explanation:
You want the area of a regular octagon, given its side length is 6 cm.
(a) Angles
A regular n-gon can be divided into n congruent isosceles triangles, each with its a.pex at the center of the polygon. The a.pex angle will have a measure of 360°/n.
The central angle of one sector of a regular octagon is 360°/8 = 45°. That means the triangle interior angle at A or B will be ...
∠OAM = (180° -45°)/2
∠OAM = 67.5°
Apothem
The apothem of a regular polygon is the distance from its center to the midpoint of one side. Here, it is the length of segment OM.
We know ∠OAM is 67.5°. The side OM of triangle OAM is related to the side AM by the tangent function:
Tan = Opposite/Adjacent
tan(67.5°) = OM/AM
Since M is the midpoint of the 6 cm length AB, the measure of AM = 3 cm. That lets us find OM:
OM = AM·tan(67.5°) = (3 cm)tan(67.5°)
OM ≈ 7.2426 cm
(b) Area
The area of the octagon will be 8 times the area of a sector triangle. This, ...
A = 8(1/2sa) . . . . . . . . . . where s is the side length, and a is the apothem
A = 4(6 cm)(7.2426 cm)
A ≈ 173.82 cm²
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Additional comment
Effectively, we have found a formula for the area of any regular n-gon using only the side length.
a = s/2·tan(90° -180°/n)
A = ns/2·a = (1/2)ns·(1/2)s·tan(90° -180°/n)
A = ns²/(4·tan(180°/n))
For this octagon, the formula gives ...
A = 8·(6 cm)²/(4·tan(22.5°)) ≈ 173.82 cm²
The formula A = (n/2)sa is often written A = 1/2Pa, where P is the perimeter (=ns).
You may notice that many regular polygon area problems give inconsistent values for the side length and apothem. Specifying one is sufficient for finding the area. Specifying numerical values for both is trouble.