Special Right Triangle Calculator
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.
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.
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
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
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
6 * (1 + 0 * sqrt(3)) - Hypotenuse
6 * (1 * sqrt(2) + 0 * 2) - Area
6 * 6 / 2 - Perimeter
6 + 6 + 8.4852814 - Hypotenuse ÷ shortest side
8.4852814 / 6
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 CalculatorFrom One Leg
- 30-60-90 Triangle CalculatorAll Sides