30 60 90 Triangle Calculator

A right triangle with the fixed side ratio 1 : √3 : 2.

How the Math Works

A 30-60-90 triangle is a special right triangle with angles of 30°, 60°, and 90° degrees. The sides of this triangle follow a fixed ratio of 1 : √3 : 2, where the shortest side (opposite the 30° angle) is half the length of the hypotenuse, and the middle side (opposite the 60° angle) is √3 times the shortest side. When you input the length of the short leg, the calculator multiplies this value by √3 to find the long leg and by 2 to find the hypotenuse, instantly solving all dimensions of the triangle using these proportional relationships.

Practical Applications

To use this calculator practically, simply enter the measured length of the shortest side when you have a 30-60-90 triangle in real-world applications such as construction, architecture, or engineering problems. The calculator will immediately output the lengths of the other two sides, eliminating the need for manual trigonometric calculations or proportional reasoning. This is especially useful when working with roof pitches, ramp designs, or any structure that incorporates these characteristic angle measurements.

Day-to-Day Use

Understanding 30-60-90 triangle relationships helps in everyday tasks like determining optimal ladder placement against a wall, calculating material needs for diagonal braces in DIY projects, or estimating distances when hiking and navigating terrain with inclined paths. The calculator makes these geometric concepts accessible without requiring memorization of formulas, empowering students, homeowners, and professionals to make accurate measurements quickly in situations involving right triangles with these specific angle combinations.

Worked example

Short 4 → long 6.93, hypotenuse 8.

FAQ