30-60-90 Triangle Calculator
With this 30-60-90 triangle calculator one known side is enough to find all the others. Example: if the short leg is 5, the long leg is 8.66025 and the hypotenuse is 10.
Example: Known side: Short leg (opposite 30°) · Length 5 → Result: Short leg a = 5 · long leg b = 8.66025 · hypotenuse c = 10. Source: Wikipedia – Special right triangle. Updated: .
How it is calculated
A 30-60-90 triangle is a right triangle with angles of 30°, 60° and 90°. Its sides are always in the fixed ratio 1 : √3 : 2, so one known side is enough.
Formulas and steps
With the short leg a, opposite the 30° angle: long leg b = a · √3 and hypotenuse c = 2a. The area is A = a² · √3 / 2 and the perimeter P = a · (3 + √3). If the long leg is given, use a = b / √3; for a given hypotenuse, a = c / 2.
Worked example
For a = 5 we get b = 5 · √3 ≈ 8.66025 and c = 10. The area is 25 · √3 / 2 ≈ 21.6506 and the perimeter 15 + 5 · √3 ≈ 23.6603. As a check, 5² + 8.66025² = 25 + 75 = 100 = 10², so Pythagoras holds.
- Where the ratio comes from: cut an equilateral triangle of side 2a along its height and you get exactly this triangle.
- The hypotenuse is always twice the short leg, a quick plausibility test.
Frequently asked questions
What is the side ratio of a 30-60-90 triangle?
Short leg : long leg : hypotenuse = 1 : √3 : 2. With a short leg of 5 that is 5 : 8.66025 : 10.
How do I find the sides from the hypotenuse?
The short leg is half the hypotenuse and the long leg is √3 times that. For c = 10, a = 5 and b ≈ 8.66025.
Which side is opposite which angle?
The shortest side is opposite the 30° angle, the middle one opposite the 60° angle and the longest, the hypotenuse, opposite the right angle.
Sources and legal basis
As of: · Editorial policy
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