3305.51 – Making a Regular Octagon

From the plywood square shown in the figure, the corners are to be cut off to make a regular octagon. The square is aa inches on a side. How many inches xx should be cut from each corner for this purpose? Find an exact value for xx and also the correct value to the nearest quarter inch when a=12"a = 12", i.e. when the plywood is a one foot square.



The diagonal sides of the octagon will be x2x \sqrt{2} (see the figure). The horizontal and vertical sides will be a2xa - 2 x. Setting them equal gives:

a2x=x2a=2x+x2=x(2+2)a - 2 x = x \sqrt{2} \leadsto a = 2 x + x \sqrt{2} = x (2 + \sqrt{2})

therefore x=a/(2+2)x = a/(2 + \sqrt{2}).

If a=12a = 12, then

x=122+2=3.514723.50. x = \frac{12}{2 + \sqrt{2}} = 3.51472\ldots \approx 3.50.

The answer is 3 and a half inches, to the nearest quarter inch.