Math CentralQuandaries & Queries


Question from Heidi, a student:

Draw an octagon with an area of 18 square centimeters


I think of a regular octagon as a square with the 4 corners lopped off.


I let the length of the side of the octagon be s cm, chopped it up as in the diagram and introduced the variable x. The octagon is then partitioned into 9 regions

1 square of area s2 cm2
4 rectangles each of area xs cm2
4 triangles

Two triangles can be put together along the hypotenuse to form a square which is x cm by x cm. Thus the total area of the regular octagon is

s2 + 4 xs + 2 x2 = 18 cm2

Each pink triangle is a right triangle with side lengths x, x and s cm. Thus by Pythagoras theorem

x2 + x2 = s2

Solve this equation for s, substitute into the equation for the area and solve for x. Once you have x you can use x2 + x2 = s2 to find s.

Construct a square with side length s + 2x cm. Measure x cm in from each corner and cut off the corners. The resulting regular octagon has an area of 18 square centimeters.


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