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.
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.
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.
What is the difference between a 45-45-90 and a 30-60-90 triangle?
A 45-45-90 triangle has two equal legs with hypotenuse = leg times sqrt(2). A 30-60-90 triangle has sides in the ratio 1 : sqrt(3) : 2, where the shortest side is opposite the 30-degree angle. Both are special right triangles with fixed side ratios.
How do I calculate the area of a triangle?
The basic formula is area = (1/2) times base times height. For a right triangle, the two legs serve as base and height. For a 45-45-90 triangle with legs of length a, the area is (1/2) times a squared.