30-60-90 Triangle Calculator
Free 306090triangle Calculator for triangle. Enter values to get step-by-step solutions with formulas and graphs. Includes formulas and worked examples.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
If short leg = x, then long leg = x√3 and hypotenuse = 2x
A 30-60-90 triangle is formed by bisecting an equilateral triangle, which locks its sides into the permanent ratio 1 : √3 : 2. The short leg sits opposite the 30° angle, the long leg (√3 times longer) sits opposite the 60° angle, and the hypotenuse is exactly twice the short leg. Knowing any one of the three sides is enough to determine the other two without trigonometric tables.
Worked Examples
Example 1: Given the short leg
Problem:A 30-60-90 triangle has a short leg of 12 cm. Find the other sides.
Solution:Short leg = 12\nLong leg = 12√3 ≈ 20.7846\nHypotenuse = 2 × 12 = 24
Result:Long leg ≈ 20.7846 cm, hypotenuse = 24 cm
Example 2: Given the hypotenuse
Problem:A ladder triangle has a hypotenuse of 30 ft in a 30-60-90 setup. Find the legs.
Solution:Short leg = 30 ÷ 2 = 15\nLong leg = 15√3 ≈ 25.9808
Result:Short leg = 15 ft, long leg ≈ 25.9808 ft
Frequently Asked Questions
What is a 30-60-90 triangle?
A 30-60-90 triangle is a special right triangle with interior angles of 30°, 60°, and 90°. Its side lengths always follow the fixed ratio 1 : √3 : 2.
What is the side ratio for a 30-60-90 triangle?
If the short leg is x, then the long leg is x√3 and the hypotenuse is 2x. That ratio is what makes this triangle easy to solve from any one side.
What is a 45-45-90 triangle?
A 45-45-90 triangle is a special right triangle with two 45-degree angles and one 90-degree angle. The two legs are equal in length, and the hypotenuse is always the leg length multiplied by the square root of 2 (approximately 1.414).
How do I find the hypotenuse of a right triangle?
Use the Pythagorean theorem: hypotenuse = square root of (a squared + b squared), where a and b are the two legs. For a 45-45-90 triangle with legs of length 5, the hypotenuse is 5 times the square root of 2, or approximately 7.07.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy