Special Right Triangles Calculator — 45-45-90 & 30-60-90 (Free)

A special right triangles calculator uses the fixed side ratios of 45-45-90 and 30-60-90 triangles to find every side from one known value. A 45-45-90 triangle has sides x : x : x√2, so a leg of 5 gives a hypotenuse of 5√2 ≈ 7.0711. A 30-60-90 triangle has sides x : x√3 : 2x, so a hypotenuse of 10 gives legs 5 and 5√3.

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How to use the special right triangles calculator

1. Pick the triangle: 45-45-90 or 30-60-90.
2. Choose what you know: a leg, the hypotenuse, the area or the perimeter (for 30-60-90 you can pick the short leg or the long leg).
3. Type the value. Whole numbers, decimals, fractions and radicals all work: 8, 2.5, 7/2, 6√2 or 3√2/2.
4. Read all three sides as exact radicals with decimals, plus the area, perimeter, height to the hypotenuse, inradius and circumradius. Open Show steps for the full working and a Pythagorean check.

The special right triangle ratios

Two right triangles come up so often in geometry and trigonometry that their side ratios are worth memorizing. Once you know one side, the ratio gives the other two without the Pythagorean theorem or trig functions.

45-45-90 triangle: x : x : x√2

A 45-45-90 triangle is an isosceles right triangle. It is exactly half of a square cut along its diagonal. The two legs are equal, and the hypotenuse is a leg times √2 (about 1.4142). Going the other way, a leg is the hypotenuse divided by √2, which you write without a radical in the denominator as hypotenuse × √2/2.

30-60-90 triangle: x : x√3 : 2x

A 30-60-90 triangle is half of an equilateral triangle. The short leg x sits opposite the 30° angle, the long leg x√3 (about 1.7321 times the short leg) sits opposite 60°, and the hypotenuse 2x is twice the short leg. The trick is to find the short leg first, then multiply.

Area and perimeter formulas

With leg x, a 45-45-90 triangle has area x²/2 and perimeter x(2 + √2). With short leg x, a 30-60-90 triangle has area x²√3/2 and perimeter x(3 + √3). The calculator runs these backwards too: if you know the area or perimeter, it solves for x first and then builds the sides.

Worked examples

Example 1: 45-45-90 triangle with hypotenuse 8

Each leg is 8/√2. Multiply top and bottom by √2: 8√2/2 = 4√2 ≈ 5.6569. The area is ½ × 4√2 × 4√2 = ½ × 32 = 16, and the perimeter is 4√2 + 4√2 + 8 = 8 + 8√2 ≈ 19.3137.

Example 2: 30-60-90 triangle with long leg 9

Find the short leg first: 9/√3 = 9√3/3 = 3√3 ≈ 5.1962. The hypotenuse is twice that, 6√3 ≈ 10.3923. The area is ½ × 3√3 × 9 = 27√3/2 ≈ 23.3827.

Example 3: 30-60-90 triangle with short leg 4

The long leg is 4√3 ≈ 6.9282 and the hypotenuse is 2 × 4 = 8. Check: 4² + (4√3)² = 16 + 48 = 64 = 8².

Example 4: 45-45-90 triangle with perimeter 12

The perimeter is x(2 + √2) = 12, so x = 12/(2 + √2). Multiply by the conjugate 2 − √2 over itself: 12(2 − √2)/2 = 12 − 6√2 ≈ 3.5147. That is each leg, and the hypotenuse is 12√2 − 12 ≈ 4.9706.

Common side lengths

Exact trig values from special right triangles

These two triangles are where the exact sine, cosine and tangent values of 30°, 45° and 60° come from. Use the unit sides 1 : 1 : √2 and 1 : √3 : 2 and take opposite/hypotenuse, adjacent/hypotenuse and opposite/adjacent.

Common mistakes to avoid

• Putting √3 on the wrong leg. In a 30-60-90 triangle the √3 side is the long leg, opposite 60°. The shortest side is always opposite the smallest angle.
• Multiplying when you should divide. From the hypotenuse of a 45-45-90 triangle you divide by √2; you do not multiply.
• Leaving a radical in the denominator. 8/√2 is correct but not finished. Rationalize to 4√2, the form most teachers and answer keys expect.
• Using the ratios on the wrong triangle. They work only when the angles really are 45-45-90 or 30-60-90. For any other right triangle, use the Pythagorean theorem or trigonometry.
• Rounding too early. Keep √2 and √3 exact until the last step, then round once.

When to work by hand and how to check an answer

On tests without a calculator you are expected to give exact answers such as 5√3, so practise the ratios by hand and use the tool to check. Two quick checks catch most errors: the hypotenuse must be the longest side, and a² + b² must equal c². The steps panel runs the Pythagorean check for every answer.

Where special right triangles show up

A square’s diagonal is a 45-45-90 hypotenuse: a 10 ft square floor has a diagonal of 10√2 ≈ 14.1421 ft. The height of an equilateral triangle with side 6 is the long leg of a 30-60-90 triangle, 3√3 ≈ 5.1962. Carpenters use the same ratios for 45° miter cuts and roof pitches, and the unit circle values in trigonometry are built from these two triangles.

Special right triangles practice quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

What are the side ratios of a 45-45-90 triangle?

The sides are in the ratio 1 : 1 : √2. Both legs are equal and the hypotenuse is a leg times √2. A triangle with legs of 7 has a hypotenuse of 7√2 ≈ 9.8995.

What are the side ratios of a 30-60-90 triangle?

The sides are in the ratio 1 : √3 : 2, for the short leg, long leg and hypotenuse. A short leg of 5 gives a long leg of 5√3 ≈ 8.6603 and a hypotenuse of 10.

Do I need the Pythagorean theorem?

No. The ratios already contain it: for 1 : √3 : 2, 1² + (√3)² = 4 = 2². The theorem is still a handy way to check an answer, and the calculator shows that check in its steps.

How do you find the hypotenuse of a 30-60-90 triangle from the long leg?

Divide the long leg by √3 to get the short leg, then double it. For a long leg of 9: 9/√3 = 3√3, so the hypotenuse is 6√3 ≈ 10.3923. In one step, hypotenuse = 2 × long leg × √3/3.

Is a 3-4-5 triangle a special right triangle?

It is a right triangle with whole-number sides (a Pythagorean triple), and some books call it special for that reason. Its angles are about 36.87° and 53.13°, not 30°, 45° or 60°, so the radical ratios on this page do not apply. Use the Pythagorean theorem or the triangle calculator for it.

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