30-60-90 Triangle Calculator
Every side of a half-equilateral triangle from the shortest one. The hypotenuse is exactly twice it and the third side √3 times it, whatever the size.
This is an equilateral triangle cut down the middle, which is why the shortest side is exactly half the hypotenuse — the cut halves one of the equal sides. The third side is √3 times the shortest.
With shortest side 6 ft, special right triangle comes to 12.0000 ft — hypotenuse. It is reached in 5 steps, the last of which is 6 * (0 * sqrt(2) + 1 * 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 * (0 + 1 * sqrt(3))10.392 ft- Hypotenuse
6 * (0 * sqrt(2) + 1 * 2)12 ft- Area
6 * 10.392305 / 231.177 sq ft- Perimeter
6 + 10.392305 + 1228.392 ft- Hypotenuse ÷ shortest side
12 / 62
Which side is which?
The shortest side always faces the 30° angle, the √3 side faces the 60°, and the hypotenuse faces the right angle. Getting those the wrong way round is the entire difficulty of this triangle; the ratios themselves never move.
Where does it turn up in practice?
Anywhere an equilateral triangle is split down the middle: hexagonal layouts, roof pitches near 30°, and the height of an equilateral triangle, which is its side times √3/2. A 10 ft equilateral triangle stands 8.66 ft tall, and that is this triangle doing the work.
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.