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 for accuracy by Manoj Kumar, Mathematics Educator
30-60-90 Triangle Calculator
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Formula: If short leg = x, then long leg = x√3 and hypotenuse = 2x
Worked example — Long leg ≈ 20.7846 cm, hypotenuse = 24 cm
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 Long leg = 12√3 ≈ 20.7846 Hypotenuse = 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 Long 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.
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.
When are special right triangles used in real life?
Special right triangles appear in construction (roof pitches, stair angles), navigation, engineering (bridge supports, truss designs), and design. A 45-degree angle creates equal rise and run, commonly used for roof slopes and diagonal bracing.
How do I know if a triangle is a right triangle?
Check if the sides satisfy the Pythagorean theorem: a squared + b squared = c squared, where c is the longest side. Common right triangle side ratios include 3:4:5, 5:12:13, 8:15:17, and 7:24:25.
How do I find a missing side of a triangle?
For right triangles, use the Pythagorean theorem. For any triangle, use the Law of Cosines (c squared = a squared + b squared - 2ab cos C) or the Law of Sines (a/sin A = b/sin B = c/sin C). Special triangles have fixed ratios that simplify calculations.
What is an isosceles right triangle?
An isosceles right triangle is the same as a 45-45-90 triangle. It has two equal legs and a right angle between them. The word isosceles means two sides of equal length. Every 45-45-90 triangle is isosceles, and every isosceles right triangle is a 45-45-90.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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