30 60 90 Triangle Calculator
Calculate the long leg, hypotenuse, perimeter, area, radii, and altitude from the short leg. 30 60 90 Triangle Calculator helps students, teachers, engineers, and builders solve special right triangle measurements quickly and accurately.
Highlights
- Calculate all key measurements of a 30-60-90 triangle from the short leg.
- Find the long leg and hypotenuse using special triangle ratios.
- Determine perimeter, area, and altitude to the hypotenuse.
- Calculate the triangle’s inradius and circumradius.
- Supports multiple length units and displays all three angle measures.
Table of Contents
How 30 60 90 Triangle Calculator Works?
A 30-60-90 triangle is a special right triangle whose angles are fixed at 30°, 60°, and 90°. If the short leg, opposite the 30° angle, is represented by x, every other measurement can be found from the standard side ratio 1 : √3 : 2.
The calculator uses the known short leg to calculate the remaining side lengths, perimeter, area, altitude to the hypotenuse, inradius, circumradius, and fixed angles.
| Output | Formula | Meaning |
|---|---|---|
| Long Leg | The longer perpendicular side | |
| Hypotenuse | The side opposite the right angle | |
| Perimeter | The sum of all three sides | |
| Area | The enclosed region | |
| Altitude to Hypotenuse | The perpendicular distance from the right angle to the hypotenuse | |
| Inradius | The radius of the inscribed circle | |
| Circumradius | The radius of the circle through all three vertices |
The perimeter is calculated by adding the short leg, long leg, and hypotenuse. The area uses one-half times the product of the two perpendicular legs. For any right triangle, the altitude to the hypotenuse is the product of the legs divided by the hypotenuse, while the circumradius is half the hypotenuse. The angles remain fixed at 30° opposite the short leg, 60° opposite the long leg, and 90° between the two legs. The calculator accepts short-leg values from 0.01 through 1,000,000,000 in the selected supported length unit and reports area in the corresponding square unit.
Practical Use Cases
Students can enter a known short leg to check homework involving special right-triangle ratios, then compare the calculated long leg, hypotenuse, area, and fixed angles with their manual work.
Teachers can use the calculator to create examples and demonstrate how the ratio 1 : √3 : 2 produces every side measurement in a 30-60-90 triangle.
Engineers can use the results for quick preliminary geometry checks involving triangular components, including perimeter, enclosed area, radii, and the altitude to the hypotenuse.
Builders can apply the special-triangle relationships when laying out right-angled supports, ramps, braces, or triangular cuts where one short-leg measurement is known.
Accuracy and Limitations
The calculator uses exact 30-60-90 relationships and evaluates √3 numerically, so results are mathematically accurate subject to normal rounding in the displayed output. The input must be a positive short-leg length between 0.01 and 1,000,000,000, and the triangle must have exactly 30°, 60°, and 90° angles; it is not suitable for arbitrary right triangles or triangles defined by different known sides. Side-related outputs use the selected supported length unit, area uses its corresponding square unit, and the angle outputs are fixed at 30°, 60°, and 90°.
Quick Examples
Example #1: Small Classroom Triangle
Calculate all measurements for a 30-60-90 triangle with a short leg of 10 centimeters.
Example #2: Construction Layout
Determine the dimensions and geometric properties of a triangle with a 2-meter short leg.
Example #3: Engineering Measurement
Find the key measurements of a 30-60-90 triangle based on a short leg measuring 18 inches.
