SOLUTION: 4.if the apothem of a regular octagon is a and its side is b, express the areas in terms of a and b.

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Question 203379: 4.if the apothem of a regular octagon is a and its side is b, express the areas in terms of a and b.
Answer by Earlsdon(6294) About Me  (Show Source):
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If you were to draw all of the diagonals in the regular octagon, you would see that the octagon would be divided into eight congruent isosceles triangles.
The base of each of these triangles is the side of the octagon, or b.
The height of each of these triangles is the apothem, or a.
The apothem is really just the radius of the inscribed circle of the octagon.
Now the area of each isosceles triangle is given by:
A+=+%281%2F2%29bh but in this problem, h = a(the apothem) and b = b(the side), so we have:
A+=+%281%2F2%29ab but since there are eight of these triangles in the entire octagon, the area of the octagon can be expressed, in terms of a and b, as:
A+=+8%281%2F2%29ab
highlight%28A+=+4ab%29