30-60-90 Triangle Calculator

Calculate all sides, area, perimeter, altitudes, and radii of a 30-60-90 special right triangle. Enter any one side and get every property from the same ratio setup.

cm
Planning notes, formulas, and examples

About the 30-60-90 Triangle Calculator

The 30-60-90 triangle is one of the two "special" right triangles in geometry (the other being the 45-45-90). Its fixed side ratios of 1: √3: 2 let you compute every dimension from a single known side — no trigonometry tables required. The short leg (opposite 30°) is half the hypotenuse, the long leg (opposite 60°) is √3 times the short leg, and the hypotenuse is twice the short leg.

This triangle appears everywhere: it is exactly half of an equilateral triangle split along its altitude. Engineers encounter it in structural trusses, roof pitches, and hexagonal geometry. Surveyors use the ratio to compute inaccessible heights, and physicists apply it when resolving forces at 30° or 60° angles.

Beyond the three sides, this calculator derives the full suite of properties: area, perimeter, semi-perimeter, all three altitudes, the circumradius (R), the inradius (r), and the medians. A visual ratio comparison bar chart and a reference table of common 30-60-90 triangles make exploring the relationships intuitive. Choose a preset or enter any single side — short leg, long leg, or hypotenuse — and see every measurement update from the same special-triangle ratios.

Knowing the 30-60-90 ratios by heart is a powerful shortcut for standardized tests (SAT, GRE, ACT) as well as practical tasks in carpentry, machining, and graphic design.

When This Page Helps

The 30-60-90 triangle appears so often that it is worth treating as a geometry shortcut instead of a full trigonometry problem. This calculator is useful when you want to move quickly from one known side to the entire triangle, including the area, altitudes, medians, and circle radii. It is especially practical for test prep, drafting, surveying setups, and any design work involving equilateral triangles, hexagons, or 60-degree layouts where the fixed 1: √3: 2 ratio saves time.

How to Use the Inputs

  1. Choose which side you know: Short Leg, Long Leg, or Hypotenuse.
  2. Select a measurement unit (mm, cm, in, m, ft).
  3. Enter the known side length.
  4. Or click a preset to load a common example.
  5. View all three sides, area, perimeter, altitudes, and radii in the output cards.
  6. Compare dimensions visually in the ratio bar chart.
  7. Scroll down for the reference table of standard 30-60-90 triangles.
Formula used
Ratios — short: long: hypotenuse = 1: √3: 2 Short leg: a Long leg: b = a√3 Hypotenuse: c = 2a Area: A = (a × b) / 2 = (a² √3) / 2 Perimeter: P = a + a√3 + 2a = a(3 + √3) Altitude to hypotenuse: h_c = (a × b) / c = (a√3) / 2 Circumradius: R = c / 2 = a Inradius: r = (a + b − c) / 2

Example Calculation

Result: Short ≈ 6.93 cm, Long = 12 cm, Hyp ≈ 13.86 cm, Area ≈ 41.57 cm²

If the known side is the long leg and value = 12 cm, the short leg is 12 / √3 ≈ 6.93 cm. The hypotenuse is 2 × 6.93 ≈ 13.86 cm. Area = (6.93 × 12) / 2 ≈ 41.57 cm², and the perimeter is about 6.93 + 12 + 13.86 = 32.78 cm.

Tips & Best Practices

  • A 30-60-90 triangle is exactly half of an equilateral triangle — draw the altitude of an equilateral triangle to see it.
  • The hypotenuse is always exactly twice the shortest side — a quick check for your answer.
  • The circumradius equals the short leg (R = a), so the circumscribed circle passes through the right-angle vertex at distance a from the center.
  • For SAT / GRE: memorize 1: √3: 2 and 1: 1: √2 for instant triangle solutions.
  • In 3D geometry, hexagonal cross sections decompose into six 30-60-90 triangles — useful for bolt patterns and honeycomb structures.

Why The 1: √3: 2 Ratio Matters

A 30-60-90 triangle is completely determined by a single side because the other two sides are locked into the ratio 1: √3: 2. That means every property downstream, including area, perimeter, inradius, circumradius, and the altitude to the hypotenuse, can be written in terms of one measurement. This is why the shape is such a standard shortcut in geometry courses and technical drawing: once you identify the triangle type, most of the work is already done.

Connection To Equilateral And Hexagonal Geometry

Splitting an equilateral triangle down the middle creates two congruent 30-60-90 triangles. The same structure appears repeatedly in regular hexagons, bolt-circle layouts, triangular trusses, and honeycomb-style designs. If you know the side of a hexagon or the altitude of an equilateral panel, you can often reduce the problem to this special triangle and solve it with simple multiplication instead of full trigonometric tables.

Fast Checks For Homework And Field Work

A quick consistency check is to compare the known values against the special-triangle pattern. The hypotenuse should be exactly twice the short leg, and the long leg should be about 1.732 times the short leg. If those relationships do not match your manual work, either the triangle is not 30-60-90 or one of the numbers was entered incorrectly. That makes this calculator useful both for solving new problems and for verifying hand calculations before you move on to a larger design or proof.

Sources & Methodology

Last updated:

Frequently Asked Questions

  • The sides are always in the ratio 1: √3: 2. The shortest side is opposite the 30° angle, the medium side is opposite 60°, and the longest side (hypotenuse) is opposite the 90° angle.

More in this topic

45-45-90 Triangle Calculator

Calculate all sides, area, perimeter, altitudes, and radii of a 45-45-90 isosceles right triangle. Enter any one side and get every property from the same triangle definition.

Right Triangle Calculator

Solve any right triangle from two known values — legs, hypotenuse, or angles. Computes all sides, angles, area, perimeter, inradius, circumradius, and altitude. Includes Pythagorean triple presets.