45-45-90 triangle calculator, one side gives the rest.

A 45-45-90 triangle calculator finds every side of an isosceles right triangle from one known value. The two legs are equal, and the hypotenuse is a leg times √2, about 1.41421. Type a leg, the hypotenuse, the area or the perimeter, and this calculator gives both legs and the hypotenuse in exact form (such as 5√2) and as decimals, the area, the perimeter, the height to the hypotenuse, the inradius and the circumradius, with a drawing to scale and the working.

How to use the 45-45-90 triangle calculator

1. Under What you know, choose A leg, The hypotenuse, The area or The perimeter.
2. Type the number in the box below it. Whole numbers, decimals and fractions such as 7/2 all work.
3. Pick a Unit if you want it shown, and the Decimal places.
4. Read the three side cards. Each shows the decimal and the exact form beneath. The drawing is to scale, and The working shows each step with your numbers.
5. Press Copy result for a one-line summary, or Copy link to share the same triangle.

Why the sides are in the ratio 1 : 1 : √2

The angles 45°, 45° and 90° mean the two legs are equal. Call each one x. Pythagoras says the hypotenuse squared is x² + x² = 2x², so the hypotenuse is x√2. That is where the ratio 1 : 1 : √2 comes from. Because √2 is irrational, the exact hypotenuse of a leg-5 triangle is 5√2, and 7.0711 is only a rounded copy of it.

Working from a leg or from the hypotenuse

You know a leg. The other leg is the same, and the hypotenuse is the leg times √2. For a leg of 5, the hypotenuse is 5 × 1.41421 = 7.0711, written exactly as 5√2. The area is ½ × 5 × 5 = 12.5, and the perimeter is 5 + 5 + 7.0711 = 17.0711.

You know the hypotenuse. Divide by √2. For a hypotenuse of 10, each leg is 10 ÷ 1.41421 = 7.0711. Exactly, that is 5√2, because 10 ÷ √2 = 10√2 ÷ 2. The area is ½ × 7.0711 × 7.0711 = 25.

You know the area. With legs L the area is L² ÷ 2, so L = √(2 × area). An area of 50 gives L = √100 = 10 and a hypotenuse of 14.1421.

You know the perimeter. The perimeter is L × (2 + √2), so L = perimeter ÷ 3.41421. A perimeter of 20 gives a leg of 5.8579 and a hypotenuse of 8.2843.

Leg and hypotenuse side by side

Where you meet a 45-45-90 triangle

Cut any square along its diagonal and you get two of them. A square tile 12 inches on a side has a diagonal of 12√2, which is 16.9706 inches. That is why a square room's corner-to-corner distance is always about 1.414 times the wall. A right-angle cut on a 45 degree mitre uses the same triangle, and so does a set square with a 45 degree angle.

Other results the calculator gives

The height to the hypotenuse is the line from the right angle straight down to the hypotenuse. In this triangle it is exactly half of the hypotenuse, so for a leg of 5 it is 5√2 ÷ 2 = 3.5355. The circumradius, the radius of the circle through all three corners, is also half of the hypotenuse, 3.5355. The inradius, the radius of the circle that just touches all three sides, is half of (leg + leg − hypotenuse), which is 1.4645 for a leg of 5.

Check it by hand

Pythagoras gives a quick test. For a leg of 5, the two legs squared add to 25 + 25 = 50. The hypotenuse 5√2 squared is 25 × 2 = 50, which matches. The decimal 7.0711 squared is 50.0000, to four places. The calculator prints this check in The working with your own numbers.

Exact form or decimal

Use the exact form when you are still calculating, because 5√2 carries no rounding error. Use the decimal when you need a length to cut, draw or write on a plan. A square of side 12 inches has a diagonal of 12√2, which is 16.9706 inches, so a rounded 17 inches would be about a third of an inch too long.

Common problems

• The decimals look slightly off when you check by hand. The calculator keeps the exact value and only rounds what it shows. Raise Decimal places to see more digits.
• You typed the hypotenuse but got a leg that is shorter. That is right. The hypotenuse is always the longest side.
• Your triangle has a 90 degree angle but the legs differ. Then it is not a 45-45-90 triangle. Use the triangle calculator instead, or the 30-60-90 triangle calculator if the angles are 30 and 60.

For a square's diagonal and area as well, try the area and volume calculator.