Question 554031
how do you to find the length of a side of a octagon with only the measurement of the total area(35cm^2)
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If it's a regular octagon, it's comprised of 8 isoceles triangles having vertex angles of 45 degs, each with an area of 35/8 sq cm.
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b = length of sides
h = height of triangle (the apothem of the octagon)
(b/2)/h = tan(22.5)
b = 2h*tan(22.5)
h = (b/2)*cot(22.5)
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b*h/2 = 35/8 (Area of 1 triangle)
{{{(b^2/2)*cot(22.5) = 35/8}}}
{{{b = sqrt(35*tan(22.5)/4)}}}
b =~ 1.904 cm
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You can use the formula for area of a polygon with n sides of length s:
{{{Area = ns^2*cot(180/n)/4}}}
Applies to all regular polygons