Eric C. answered • 11/27/15
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Engineer, Surfer Dude, Football Player, USC Alum, Math Aficionado
So what you have here is an octagon inside a circle inside a square, and all you know is that the side length of one of your octagons is 8 inches. Our job is to find the radius of our octagon, which will let us know the diameter of the circle, which will be the length of one of our squares.
Let's work our way from the inside out.
A regular octagon with a side length of 8 inches can be thought of as 8 isosceles triangles all of base length 8. We know that the sum of all interior angles of a given shape is
(n-2)(180)
where "n" is the number of sides. An octagon has 8 sides, so the sum of all interiors is (8-2)*180 = 1080 deg.
The term "regular" tells us all angles are the same, so 1 angle of the octagon is 1080/8 = 135 deg.
If we divvy up the octagon into 8 isosceles triangles, the base angles will each be 135/2 deg and the base length will be 8. We can draw a line down the middle of one of these triangles to make it into a right triangle with a base length of 4. Using the known base length and the known angle, we can figure out the hypotenuse (which, conveniently enough, will be our circle's radius).
cos(135/2) = 4/ hypotenuse.
Using the cosine half angle formula:
cos(135/2) = √[(1+cos(135))/2]
cos(135) = -cos(45) = -sqrt(2)/2
cos(135/2) = √[(1-sqrt(2)/2)/2] = √[(2-sqrt(2))/4] = 1/2*√(2-sqrt(2))
hypotenuse = 4/(1/2*√(2-sqrt(2)))
= 8/√(2-sqrt(2))
This hypotenuse is also the radius of the circle in which it's inscribed. This means the diameter of the circle is:
D = 16/√(2-sqrt(2))
Since the circle is inscribed in a square, the diameter of the circle is the side length of the square.
There are 4 rows of 3 squares each, meaning the length of the quilt is 4 squares and the width is 3 squares.
L = 4*D = 64/√(2-sqrt(2))
W = 3*D = 48/√(2-sqrt(2))
Kinda ugly, but there it is. Hope this helps.
