Triangle 30 60 90 Calculator

Calculate all side lengths of a 30-60-90 special right triangle from any one known side using fixed ratio relationships.

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Result

Hypotenuse

10

Short leg

5

Long leg

8.66

Area

21.65

Perimeter

23.66

Angle A

30°

Angle B

60°

Angle C

90°

Ratios Short : Long : Hypotenuse = 1 : √3 : 2
5 : 8.66 : 10
Area = ½ × 5 × 8.66 = 21.65

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Triangle Geometry

30-60-90 triangle: side ratios, formulas, and worked examples

A 30-60-90 triangle is a special right triangle whose angles are always 30, 60, and 90 degrees. Because the angle measures are fixed, the side lengths follow a constant ratio of 1 : sqrt(3) : 2, which means knowing any one side is enough to find the other two.

The fixed side ratio

In every 30-60-90 triangle the shortest side is opposite the 30-degree angle, the medium side is opposite the 60-degree angle, and the hypotenuse is opposite the right angle. If the shortest side has length a, then the medium side is a * sqrt(3) and the hypotenuse is 2a.

This ratio is derived from bisecting an equilateral triangle along its altitude. The altitude splits the equilateral triangle into two congruent 30-60-90 triangles, each sharing the same proportional relationship.

Core formulas

Once you know any single side, the remaining sides and the area follow directly from the fixed ratio.

short = hypotenuse / 2

The side opposite 30 degrees is half the hypotenuse.

medium = short * sqrt(3)

The side opposite 60 degrees is the short side times the square root of three.

Area = (short * medium) / 2

Standard right-triangle area using the two legs.

When to use this triangle

The 30-60-90 triangle appears frequently in construction, drafting, and trigonometry problem sets. Roof pitches, ramp grades, and hexagonal tile layouts all rely on the 30-60-90 relationship because the ratio produces clean geometric proportions.

Frequently asked questions

If the hypotenuse is 10, what are the other sides?

The short side is 5 and the medium side is 5 * sqrt(3), which is approximately 8.66.

 How is a 30-60-90 triangle related to an equilateral triangle?

Cutting an equilateral triangle in half along its altitude produces two congruent 30-60-90 triangles.

 Can a 30-60-90 triangle be isosceles?

No. All three sides have different lengths because all three angles are different.

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