An octagon is a geometric figure that lies on a plane and has eight sides. The term typically refers to a regular, simple octagon unless otherwise specified, meaning that its sides are of equal length and do not intersect. The standard stop sign is in the shape of a regular, simple octagon. The length of the sides of this type of octagon may be calculated when its area is given.
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- Moderately Easy
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Inscribe an octagon that has sides of length "a" within a square that has sides of length "S." Note that the area of the square is equal to the area of the inscribed octagon plus the area of the four "corners" of the square.
Find the angles of the triangle formed by the corners of the square described in Step 2. Each of these corners is a right triangle, since one angle is shared with the square. The other two angles form external angles of the octagon and are therefore equal, since the octagon is a regular, simple polygon.
Determine the length of the legs "b" of each corner triangle. Since the angles of these right triangles are equal, the lengths of their legs are also equal. We therefore have a^2 = b^2 +b^2 = 2b^2 by the Pythagorean Theorem, and 2b^2 = a^2 gives us b^2 = a^2/2 so b = a/(2^(1/2)).
Calculate the area of the corner triangles. Using the formula for the area of a triangle, we have 1/2 x b x b = 1/2 x b^2 = 1/2 x (a/(2^(1/2)))^2 = 1/2 (a^2)/2 = 1/2 a^2/2 = a^2/4. The total area of these four triangles is therefore 4 x a^2/4 = a^2.
Calculate the area of the square. Each side of the square has length a^2/2^(1/2) + a + a^2/2^(1/2) = a + 2 x a^2/2^(1/2) = a + 2^(1/2) x a = a(1 + 2^(1/2)). The area of the square is therefore (a(1 + 2^(1/2)))^2 = a^2 x (1 + 2^(1/2))^2 = a^2 x (1 + 2 x 2^(1/2) + 2) = a^2 x (3 + 2 x 2^(1/2)) = 3a^2 + 2a^2 x 2^(1/2).
Subtract the area of the corner triangles from the area of the square to get the area A of the octagon. We have A = (3a^2 + 2a^2 x 2^(1/2)) - a^2 = 2a^2 + 2a^2 x 2^(1/2)) = 2a^2(1 + 2^(1/2), so A = 2a^2(1 + 2^(1/2)).
Solve for the length of side a in terms of area "A." A = 2a^2(1 + 2^(1/2)) implies A/(2(1+2^(1/2)) = a^2, which means a = (A/(2(1+2^(1/2)))^(1/2).
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