30-60-90 Triangle Calculator
From any one measurement of a 30-60-90 triangle — short leg, long leg, hypotenuse, height, area or perimeter — get all three sides in the 1 : √3 : 2 ratio as exact radicals and decimals, plus the area, perimeter, height, inradius, circumradius, angles and step-by-step working.
How it works
short : long : hypotenuse = 1 : √3 : 2 · long = short × √3 · hypotenuse = 2 × short · Area = ½ · short · long = (√3 ÷ 4) · hypotenuse² · Perimeter = short × (3 + √3) · Height to hypotenuse = short × √3 ÷ 2 · Inradius = short × (√3 − 1) ÷ 2 · Circumradius = hypotenuse ÷ 2
A 30-60-90 triangle is a right triangle whose angles fix its shape, so all three sides scale together in the exact ratio 1 : √3 : 2 — short leg, long leg, hypotenuse. Knowing any one measurement pins the scale factor: find the short leg s first, then multiply by √3 for the long leg and by 2 for the hypotenuse. From the long leg, divide by √3 (which rationalises to multiplying by √3/3); from the height to the hypotenuse, s = 2h ÷ √3; from the perimeter, s = P ÷ (3 + √3); from the area, s = √(2A ÷ √3). The calculator keeps the answers exact where it can — entering a long leg of 6 gives a short leg of 2√3 and a hypotenuse of 4√3, not just 3.4641 and 6.9282 — and shows the decimals to the precision you choose. It also gives the height to the hypotenuse, the inradius and circumradius, the angles in degrees and radians, the sine, cosine and tangent of 30° and 60°, and the full working, ending with a Pythagorean check.
Worked example
Suppose the long leg is 6. The short leg is 6 ÷ √3 = 2√3 ≈ 3.4641 and the hypotenuse is twice that, 4√3 ≈ 6.9282. The area is ½ × 2√3 × 6 = 6√3 ≈ 10.3923 and the perimeter is 2√3 + 6 + 4√3 = 6 + 6√3 ≈ 16.3923. The height to the hypotenuse is 2√3 × √3 ÷ 2 = 3. Check: (2√3)² + 6² = 12 + 36 = 48 = (4√3)².
Sources: Special right triangle — Wikipedia
Frequently asked questions
Why are the sides always 1 : √3 : 2?
The angles 30°, 60° and 90° fix the triangle’s shape up to size. Dropping a perpendicular in an equilateral triangle produces exactly this half, and the Pythagorean theorem gives the long leg as √3 times the short leg with the hypotenuse twice the short leg.
Which side is the longest?
The hypotenuse, opposite the 90° angle, is always the longest at 2 units. The long leg opposite 60° is √3 ≈ 1.732 units, and the short leg opposite 30° is the shortest at 1 unit.
How do I find the short leg from the long leg?
Divide the long leg by √3. To write it without a root in the denominator, multiply top and bottom by √3: a long leg of 5 gives a short leg of 5/√3 = 5√3/3 ≈ 2.8868, and the hypotenuse is twice that, 10√3/3 ≈ 5.7735.
Can I start from the area or perimeter?
Yes. Pick Area or Perimeter under “What do you know?”. The perimeter is s(3 + √3), so s = P ÷ (3 + √3) and the answers stay exact. The area is (√3 ÷ 2)s², so s = √(2A ÷ √3), which is usually not a simple radical, so decimals are shown.
What are sin 30°, cos 30° and tan 30°?
Read them from the 1 : √3 : 2 sides: sin 30° = 1/2, cos 30° = √3/2 and tan 30° = 1/√3 = √3/3. For 60° they swap: sin 60° = √3/2, cos 60° = 1/2 and tan 60° = √3.
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This calculator is for educational purposes. Double-check important results before acting on them.