SOLUTION: Find the area of a regular octagon inscribed in a circle woth radius r Hint: A regular octagon consists of eight isosceles triangles that have the same shape and size

Algebra ->  Test -> SOLUTION: Find the area of a regular octagon inscribed in a circle woth radius r Hint: A regular octagon consists of eight isosceles triangles that have the same shape and size      Log On


   

Question 1120973: Find the area of a regular octagon inscribed in a circle woth radius r
Hint: A regular octagon consists of eight isosceles triangles that have the same shape and size
Found 2 solutions by Alan3354, Boreal:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Find the area of a regular octagon inscribed in a circle with radius r
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For any regular polygon:
Area+=+n%2Ar%5E2%2Atan%28180%2Fn%29 --- n = # of sides, r = radius
= r%5E2%2Atan%2822.5%29
= ~ 0.414%2Ar%5E2
= %28sqrt%282%29-1%29%2Ar%5E2
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Hint: we don't need hints.

Answer by Boreal(15235) About Me  (Show Source):
You can put this solution on YOUR website!
360/8 means the central angle of each triangle is 45 degrees. With the radius on each side being r, the other two angles are equal and each of the 8 triangles is isosceles, with angles 67.5 degrees.
The triangle can be split into two right triangles, each with angles 22.5, 67.5 and 90 degrees. The dividing line has length cos 22.5=x/r so length is r cos 22.5 or 0.9239 r.
Half the length is the base of the right triangle, whose length is r sin 22.5 or 0.3827 r. The whole base of one of the eight triangles is twice that or 0.7654.
The area of each of the 8 triangles =(1/2)bh, or 0.7654*0.9239*.5*r^2 or 0.3536 r^2
Eight of them, the answer, is 2.83 r^2