Question 249584: Find the area of a regular octagon inscribed into a circle of radius 1.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! octagon has 8 sides.
this means it has 8 angles.
circle has 360 degrees.
the octagon forms 8 triangles intersecting at the center of this circle.
divide 360 by 8 and you get 45 degrees for the central angle of each of these triangles.
we will work with one of these triangles to get the area of it.
then we will multiply the area of that triangle by 8 to get the total area for all 8 triangles which will be equal to the area of the octagon.
the radius of the circle is equal to the sides of each of these triangles.
select the following link to see a picture of the octagon and a blowup of one of the triangles that we will work to get the area of.
http://theo.x10hosting.com/problems/249584.html
the triangle we will be working with is triangle 4 in the picture.
we label it triangle ABC.
we drop a perpendicular to intersect with BC at D.
triangles ABD and ACD are congruent by SAS.
we will work with triangle ACD.
cosine (22.5) = AD / 1
solve for AD to get AD = 1 * cosine (22.5) = .923879533
this equals the height of the triangle.
sine (22.5) = DC / 1
solve for DC to get DC = 1 * sine (22.5) = .382683432
this equals the base of the triangle.
area of triangle ADC = 1/2 * b * h = 1/2 * .382683432 * .923879533 = .176776695
area of triangle ABC = area of triangle ABD plus triangle ACD = 2 * .176776695 = .353553391
there are 8 of triangle congruent to triangle ABC in the octagon, so the area of the octagon = 8 * .353553391 = 2.828427125
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