How to calculate the distance a square corner adds to the diameter of a circle? What is the formula for finding the distance a square corner adds to the diameter of a circle?
In the drawing, if I know the diameter (5.0") of the circle, how much do I add to find the total distance of its furthest point from the corner?
 A: Okay, I figured it out. 5" x square root of 2 will give the diagonal center-line of a square that tightly encompasses a circle of 5" diameter. Then, simply subtract the 5-inch diameter of the circle and divide the remaining portion by two to get the extra length of just one corner. Add the corner length back to the circle's diameter to arrive at 6.0355 inches.
The formula for the square's diagonal comes from the Pythagorean formula. Because a diagonal of a square would create mirrored right triangles, each with 45-degree corners, the diagonal is their hypotenuse. Using the Pythagorean formula, a^2 + b^2 = c^2, with a and b each lengths equal to the diameter of a circle that would fit inside, the formula for a square's diagonal is derived.
Since I want "usable" fractions that can be found on an American tape measure, with +/-16th of an inch being an acceptable tolerance, I take an extra step of multiplying the decimal fraction by 16, which gives me a half a sixteenth. Not enough to matter, so, my answer is approximately 6 inches.
