Question 249584
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.


<a href = "http://theo.x10hosting.com/problems/249584.html" target = "_blank">http://theo.x10hosting.com/problems/249584.html</a>


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