45-45-90 Triangle Calculator
Every side of an isosceles right triangle from a single leg. The hypotenuse is always √2 times a leg, at every size, with no trigonometry involved.
This is half a square cut corner to corner, which is why the two legs are equal and the hypotenuse is exactly √2 times either. The ratio holds at every size, so one measurement gives all three sides with no trigonometry.
With shortest side 6 ft, special right triangle comes to 8.4853 ft — hypotenuse. It is reached in 5 steps, the last of which is 6 * (1 * sqrt(2) + 0 * 2), and each one is printed on the page with its numbers filled in. The formula is the one published by Wolfram MathWorld, not an approximation fitted to it.
Formula and sources checked · How we check
- Second side
6 * (1 + 0 * sqrt(3))6 ft- Hypotenuse
6 * (1 * sqrt(2) + 0 * 2)8.485 ft- Area
6 * 6 / 218 sq ft- Perimeter
6 + 6 + 8.485281420.485 ft- Hypotenuse ÷ shortest side
8.4852814 / 61.414
Where does this triangle come from?
A square cut corner to corner. That is why the legs are equal, and why the diagonal of any square is its side times √2 — the same fact said twice. A 4 ft square panel has a 5.657 ft diagonal, which is the figure you check a finished frame against.
How do you go from the hypotenuse back to a leg?
Divide by √2, or multiply by 0.7071, which is the same operation. A 10 ft brace across a corner meets each wall 7.071 ft out from it. Squaring a frame with a tape measure is this arithmetic and nothing more.
The formula
- Second side
shortest * (kind_isosceles + kind_half_equilateral * sqrt(3)) - Hypotenuse
shortest * (kind_isosceles * sqrt(2) + kind_half_equilateral * 2) - Area
shortest * side_b / 2 - Perimeter
shortest + side_b + hypotenuse - Hypotenuse ÷ shortest side
hypotenuse / shortest
Formula from Wolfram MathWorld — special right triangles, NIST Digital Library of Mathematical Functions.
The same calculator, other cases
Undecided? The Special Right Triangle Calculator carries every case behind one control.