Right Triangle Calculator

How this right triangle calculator works

A right triangle has one 90° angle. The two sides that meet at that angle are the legs, a and b, and the longest side across from it is the hypotenuse, c. Angle α sits across from leg a and angle β across from leg b, so α + β = 90°. The height h runs from the right angle straight down to the hypotenuse.

Enter any two of the eight values (two sides, a side and an angle, the area and the perimeter, and so on) and this right triangle solver finds the rest: all three sides, both angles in degrees and radians, the height, area, perimeter, and the radii of the inscribed and circumscribed circles. It works as a right angle triangle calculator, a hypotenuse calculator, or a leg calculator, and it can show the calculation steps.

For example, a leg of 3 and a hypotenuse of 5 give the other leg b = √(5² − 3²) = 4, angles of 36.87° and 53.13°, an area of 6, and a perimeter of 12.

How to find the hypotenuse

With both legs known, use the Pythagorean theorem: square each leg, add them, and take the square root.

c = √(a² + b²)

With one leg and one angle, use trigonometry instead: c = a ÷ sin α, or c = a ÷ cos β. A ladder whose foot is 4 ft from the wall and whose top reaches 12 ft up the wall is √(4² + 12²) ≈ 12.65 ft long.

Right triangle formulas

• Missing leg: b = √(c² − a²)
• Angles: sin α = a ÷ c, cos α = b ÷ c, tan α = a ÷ b, and β = 90° − α
• Area: A = a × b ÷ 2, which also equals c × h ÷ 2
• Height to the hypotenuse: h = a × b ÷ c
• Perimeter: P = a + b + c
• Inradius: r = (a + b − c) ÷ 2
• Circumradius: R = c ÷ 2, since the hypotenuse is a diameter of the circumscribed circle

Special right triangles

A 45-45-90 triangle has equal legs, and its hypotenuse is the leg times √2. A 30-60-90 triangle has sides in the ratio 1 : √3 : 2, so the hypotenuse is twice the shortest leg. Right triangles whose sides are all whole numbers, such as 3-4-5, 5-12-13, and 8-15-17, are called Pythagorean triples.

Frequently asked questions

Why are there two solutions?

Some pairs of values, such as the hypotenuse and the area, do not say which leg is which. The calculator then shows both solutions, the same triangle with legs a and b swapped. When the two legs are equal there is only one.

Can I solve a right triangle from two angles?

No. Two angles fix the shape but not the size, so any number of triangles fit. Enter at least one length, such as a side, the height, the area, or the perimeter.

How do I enter angles in radians?

Choose Radians as the angle unit. You can type a decimal such as 0.5236 or a multiple of pi such as pi/6 or 2pi/5.

What units does the right triangle calculator use?

Any unit works, as long as every length uses the same one. If the sides are in feet, the area comes out in square feet; in centimeters, it comes out in square centimeters.

Limitations

Results are rounded to six significant digits, and seconds of arc to the nearest second. Values that come from measurements carry their measurement error into every result, so round to the precision you actually measured.