SOLUTION: Isosceles right triangles are cut from the four corners of a square piece of paper, 12 inches by 12 inches, so that a regular octagon is produced. What is the length, in inches,

We begin with a 12 in x 12 in square:



Then we cut out four isosceles triangles from the corners, to
produce a regular octagon (stop sign):



Let x represent the length of the legs of these isosceles
right triangles:



Next we calculate the length of the hypotenuses of all
the right triangles using the Pythagorean theorem:







So we label all these hypotenuses as 



Since the octagon is regular, all its sides have the same measure,
so we label the top, bottom, right and left sides of the octagon
also as :




Now we add up the three parts of the length of each side of the square
and set the sum equal to 12 inches, which is given as the side of the square:




Factor x out of the left side:





That's the answer, but your teacher may have intended you to
rationalize the denominator:







, or about 3.514718626 inches.

Edwin