Special Right Triangle Calculator
The 45-45-90 and 30-60-90 triangles from one known side, with every other side given as an exact multiple rather than a decimal that hides the ratio.
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 shortest * (kind_isosceles * sqrt(2) + kind_half_equilateral * 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
Triangle 45-45-90 — half a square, Shortest side 6 ft
8.4853 ft
Hypotenuse for the example below. Editing a field recomputes the calculator below; this figure holds the answer the page was loaded with.
It is written into the HTML rather than drawn by a script, so a search engine reading this page without running JavaScript still finds an answer.
- Second side
shortest * (kind_isosceles + kind_half_equilateral * sqrt(3))6 ft- Hypotenuse
shortest * (kind_isosceles * sqrt(2) + kind_half_equilateral * 2)8.485 ft- Area
shortest * side_b / 218 sq ft- Perimeter
shortest + side_b + hypotenuse20.485 ft- Hypotenuse ÷ shortest side
hypotenuse / shortest1.414
Worked example
A 45-45-90 with 6 ft legs has a hypotenuse of 6√2 — 8.4853 ft. The ratio never changes: whatever the leg, the hypotenuse is 1.41421 times it, which is why this triangle is exactly half a square cut corner to corner.
How to work it out yourself
- 1.Pick which triangle you have: two equal angles means 45-45-90, a 30° angle means 30-60-90.
- 2.Measure the shortest side — a leg on the 45-45-90, the side facing the 30° corner on the other.
- 3.For 45-45-90 the other leg is the same and the hypotenuse is √2 times it.
- 4.For 30-60-90 the second side is √3 times the shortest and the hypotenuse exactly twice it.
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
Source: Wolfram MathWorld — special right triangles, NIST Digital Library of Mathematical Functions
Questions people actually ask
- Why are these two worth memorising?
- Their side ratios are fixed, so one measurement gives you all three sides with no trigonometry. 45-45-90 is 1 : 1 : √2 and 30-60-90 is 1 : √3 : 2, and those hold at any size.
- Where does 30-60-90 come from?
- It is an equilateral triangle cut down the middle. That is why the shortest side is exactly half the hypotenuse — the cut halves one of the equal sides.
Related
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<iframe src="https://rulecalculators.com/embed/special-right-triangle" width="100%" height="420" style="border:1px solid #e7e4de;border-radius:12px" title="Special Right Triangle Calculator"></iframe>Straight to your case
The control above covers every case. These are the same calculator fixed to one of them, with the formula for that case written out.
- 45-45-90 Triangle CalculatorEvery side of an isosceles right triangle from one leg. The hypotenuse is always √2 times a leg, whatever the size.
- 30-60-90 Triangle CalculatorEvery side of a half-equilateral triangle from the shortest one. The hypotenuse is exactly twice it, and the third side √3 times it.