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30-60-90 Triangle Calculator

Calculate all sides of a 30-60-90 special right triangle from any one known side. Enter the shortest side, longer leg, or hypotenuse to find all three sides using the fixed ratio 1:sqrt(3):2, fundamental for trigonometry, geometry, and standardized testing.

Reviewed by Christopher FloiedPublished Updated

This free online 30-60-90 triangle calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Minimum: 0.01

The side opposite the 30-degree angle

Results

Longer Leg (opposite 60 deg)

8.6603

Hypotenuse (opposite 90 deg)

10

Area

21.6506

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the 30-60-90 Triangle Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The 30-60-90 Triangle Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the 30-60-90 Triangle Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

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About 30-60-90 Triangle Calculator

The 30-60-90 Triangle Calculator computes all three sides of the special 30-60-90 right triangle from the shortest side. This triangle has angles of 30, 60, and 90 degrees with sides in the exact ratio 1 : sqrt(3) : 2. It is one of the two special right triangles (the other being 45-45-90) that students must memorize for trigonometry, standardized tests like the SAT and GRE, and practical applications. The 30-60-90 triangle arises naturally when you bisect an equilateral triangle along its height. It appears in architecture (roof pitches), engineering (60-degree bracing), crystallography (hexagonal lattices), and navigation (computing components of 30-degree or 60-degree headings).

The Math Behind It

The 30-60-90 triangle is produced by cutting an equilateral triangle in half along its altitude. If the equilateral triangle has side length 2s, the altitude divides it into two congruent right triangles, each with hypotenuse 2s, short leg s (half the base), and long leg s*sqrt(3) (the altitude, by the Pythagorean theorem: (2s)^2 - s^2 = 3s^2, so the altitude is s*sqrt(3)). The exact trigonometric values for 30 and 60 degrees follow from this triangle: sin(30) = 1/2, cos(30) = sqrt(3)/2, tan(30) = 1/sqrt(3) = sqrt(3)/3; sin(60) = sqrt(3)/2, cos(60) = 1/2, tan(60) = sqrt(3). These exact values are used throughout mathematics, physics, and engineering. The 30-60-90 triangle appears in regular hexagons: each internal triangle formed by two radii and one side is equilateral, and bisecting it gives a 30-60-90 triangle. This is why hexagonal tiling uses these proportions. In crystallography, the hexagonal crystal system is based on 60-degree angles, making 30-60-90 triangles fundamental to understanding crystal structures. In structural engineering, 60-degree angles provide excellent load distribution. Equilateral triangles and their half-components (30-60-90 triangles) are inherently rigid, making them ideal for truss designs. The relationship between the sides also appears in vector decomposition: a force at 30 degrees to horizontal has components that follow the 1 : sqrt(3) ratio.

Formula Reference

30-60-90 Side Ratios

short : long : hypotenuse = 1 : sqrt(3) : 2

Variables: short = side opposite 30 deg, long = side opposite 60 deg

Worked Examples

Example 1: Short Side = 5

The shortest side of a 30-60-90 triangle is 5 units.

Step 1:Longer leg = 5 * sqrt(3) = 5 * 1.7321 = 8.6603
Step 2:Hypotenuse = 5 * 2 = 10
Step 3:Area = (5 * 8.6603) / 2 = 21.651

Sides: 5, 8.66, 10. Area = 21.65 square units.

Example 2: Equilateral Triangle Height

An equilateral triangle has sides of 10. Find its height (which forms a 30-60-90 triangle).

Step 1:Halving the equilateral triangle: short side = 10/2 = 5
Step 2:Height (long leg) = 5 * sqrt(3) = 8.6603

The height of the equilateral triangle is 8.66 units.

Common Mistakes & Tips

  • !Mixing up which side is opposite which angle. The shortest side is opposite 30 degrees, the longer leg is opposite 60 degrees, and the hypotenuse is opposite 90 degrees.
  • !Using sqrt(2) instead of sqrt(3). The 30-60-90 triangle uses sqrt(3); the 45-45-90 triangle uses sqrt(2). Do not confuse them.
  • !Forgetting that the hypotenuse is exactly twice the shortest side, not twice the longer leg.

Related Concepts

45-45-90 Triangle

The other special right triangle with ratio 1:1:sqrt(2), where both legs are equal.

Right Triangle Trig

The general right triangle solver that handles any right triangle, not just special ones.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

45-45-90 Triangle Calculator

Right Triangle Trigonometry Calculator

Frequently Asked Questions

Why is the 30-60-90 triangle so important?

It provides exact values for sin, cos, and tan of 30 and 60 degrees without a calculator. These values (1/2, sqrt(3)/2, sqrt(3)) appear constantly in mathematics, physics, and engineering. It is also the basis for understanding equilateral triangles and hexagonal geometry.

How do I find the shortest side if I know the hypotenuse?

Divide the hypotenuse by 2. If the hypotenuse is 12, the shortest side is 6, and the longer leg is 6*sqrt(3) = 10.39.

Where does the 30-60-90 triangle appear in nature?

In honeycomb structures (hexagonal cells are made of 30-60-90 triangles), in crystal lattices (hexagonal symmetry), in snowflakes (six-fold symmetry), and in basalt column formations (Giant's Causeway). The 60-degree angle is a natural consequence of optimal packing.

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