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Right Triangle Calculator

Find the area, sides, and angles of any right triangle. Enter any 2 known values and get every result, with the full step-by-step solution shown.

Right triangle calculator: enter any 2 values

This free right triangle calculator solves any right triangle from just two known values. Enter any two of the legs, hypotenuse, angles, altitude, area, or perimeter, and it instantly returns every side, both acute angles, the area, and more. It starts pre-loaded with the classic 3-4-5 triangle.

Enter any two known values

Fill in exactly two fields below (a pinned value counts as one). Everything else is solved automatically.

Angle mode

Results

3-4-5 Pythagorean triple
Leg a
3 mm
Leg b
4 mm
Hypotenuse c
5 mm
Angle α
36.8699°
Angle β
53.1301°
Altitude h
2.4 mm
Area
6 mm²
Perimeter
12 mm
Inradius
1 mm
Circumradius
2.5 mm
Right triangle calculator diagram: 3-4-5 right triangle A right triangle with legs 3 and 4 millimeters, hypotenuse 5 millimeters, area 6 square millimeters, and acute angles 36.87° and 53.13°. A B C a = 3 mm b = 4 mm c = 5 mm α = 36.87° β = 53.13°

Step-by-step solution

  1. Apply the Pythagorean theorem
    c = √(a² + b²)
    c = √(3² + 4²)
    = 5 mm
  2. Find the acute angles
    α = atan(a / b), β = 90° − α
    atan(3 / 4)
    = α = 36.8699°, β = 53.1301°
  3. Compute the area
    Area = (1/2)·a·b
    Area = (1/2)·3·4
    = 6 mm²
  4. Compute the perimeter
    P = a + b + c
    P = 3 + 4 + 5
    = 12 mm
  5. Compute the altitude to the hypotenuse
    h = (a·b) / c
    h = (3·4) / 5
    = 2.4 mm
  6. Inradius (radius of the inscribed circle)
    r = (a + b − c) / 2
    r = (3 + 4 − 5) / 2
    = 1 mm
  7. Circumradius (radius of the circumscribed circle)
    R = c / 2
    R = 5 / 2
    = 2.5 mm

How to use the right triangle calculator

  1. Type any two known values into the form: two sides, a side and an angle, or a combination with the area, perimeter, or altitude.
  2. Pick your unit (mm, cm, m, in, or ft) and choose whether angles are in degrees or radians.
  3. Read the results: all three sides, both acute angles, the area, perimeter, altitude, inradius, and circumradius.
  4. Check the step-by-step solution and the scaled diagram, then copy the shareable link to save or send your answer.

Results update as you type, and Pythagorean triples and special right triangles are labeled automatically.

Area of a right triangle calculator

Finding the area of a right triangle is easier than for any other triangle, because the two legs are already perpendicular. One leg is the base and the other is the height, so the formula is simply:

Area = ½ × a × b

For example, a right triangle with legs of 6 cm and 8 cm has an area of ½ × 6 × 8 = 24 cm². Area is always measured in square units (cm², m², in², or ft²), and the calculator converts it for you when you switch units.

Area of a right triangle without both legs

  • Hypotenuse and one leg: find the other leg with b = √(c² − a²), then use ½ × a × b.
  • Hypotenuse and one angle: Area = ½ × c² × sin α × cos α.
  • One leg and one angle: b = a / tan α, so Area = a² / (2 × tan α).
  • Hypotenuse and altitude: Area = ½ × c × h, where h is the height drawn to the hypotenuse.

You don't need to memorize these. Enter what you know into the area of a right triangle calculator above and it picks the right formula. It also works in reverse: enter the area and one side to find the missing side. See also the right triangle formulas reference.

Right triangle calculator with angles

The calculator works just as well with angles as with sides. When you know one acute angle and one side, the trigonometric ratios (SOH-CAH-TOA) give you everything else:

  • sin α = opposite / hypotenuse = a / c
  • cos α = adjacent / hypotenuse = b / c
  • tan α = opposite / adjacent = a / b

To find the angles from two sides, the calculator runs those ratios backwards: α = arctan(a / b), or α = arcsin(a / c) when you know the hypotenuse. The second angle is then β = 90° − α, because the two acute angles of every right triangle add up to 90°.

Angles can be in degrees or radians. An angle alone can't fix a triangle's size, so pair it with a side, the area, or the perimeter.

What is a right triangle?

A right triangle (or right-angled triangle) is a triangle with one angle of exactly 90°. The side opposite the right angle is the hypotenuse, and it is always the longest side. The two sides that form the right angle are the legs. Because a triangle's angles add up to 180°, the other two angles must add up to 90°.

Here, legs a and b sit opposite angles α and β, and c is the hypotenuse. The sides are linked by the Pythagorean theorem, a² + b² = c². If you only need a missing side, use the Pythagorean theorem calculator.

Worked example: solving a 3-4-5 right triangle

The calculator above starts with the most famous right triangle: legs of 3 mm and 4 mm. The hypotenuse is c = √(3² + 4²) = √25 = 5 mm. The area is ½ × 3 × 4 = 6 mm², and the perimeter is 3 + 4 + 5 = 12 mm. The angles are α = arctan(3/4) ≈ 36.87° and β = 90° − 36.87° = 53.13°. The altitude to the hypotenuse is h = (3 × 4)/5 = 2.4 mm, the inradius is (3 + 4 − 5)/2 = 1 mm, and the circumradius is 5/2 = 2.5 mm. Because 3-4-5 is a Pythagorean triple (three whole numbers with a² + b² = c²), the calculator flags it, as it does for scaled copies like 6-8-10.

Special right triangles

Two right triangles have fixed side ratios, so you can solve them from a single side. The 45-45-90 triangle is an isosceles right triangle with sides in the ratio 1 : 1 : √2. The 30-60-90 triangle has sides in the ratio 1 : √3 : 2. Use the special right triangle calculator to solve either one, or go straight to the 30-60-90 triangle calculator or the isosceles right triangle calculator.

Key definitions

Leg
One of the two shorter sides that form the right angle, called a and b here.
Hypotenuse
The side opposite the right angle and the longest side of a right triangle, called c here.
Altitude to the hypotenuse
The perpendicular line from the right angle to the hypotenuse. Its length is (a × b)/c.

Right triangle calculator FAQ

How do you find the area of a right triangle?

Multiply the two legs and divide by 2: Area = ½ × a × b. The legs are the two sides that meet at the right angle. If you only know the hypotenuse and one leg, find the missing leg first with the Pythagorean theorem, b = √(c² − a²).

Can I find the area of a right triangle from the hypotenuse alone?

No. A single hypotenuse fits infinitely many right triangles with different areas, so you need one more value: a leg, an angle, the altitude, or the perimeter. The largest possible area for a hypotenuse c is c²/4, reached when the triangle is a 45-45-90 triangle.

How do I find the angles of a right triangle?

Use an inverse trig function on any two sides: α = arctan(a/b), α = arcsin(a/c), or α = arccos(b/c). The other acute angle is 90° − α. Enter the two sides in the calculator and it shows both angles with the working.

How do you calculate a right triangle?

Start from any two known values, at least one of them a length. With two sides, use the Pythagorean theorem a² + b² = c² for the third side and arctan(a/b) for an angle. With one side and one angle, use sine, cosine, or tangent. The area is ½ × a × b and the perimeter is a + b + c. The full list is on the right triangle formulas page.

What are the rules for right triangles?

A right triangle, also called a right-angled triangle, has exactly one 90° angle, and all three angles add up to 180°, so the other two are acute and sum to 90°. The hypotenuse is the side opposite the right angle and is always the longest side; the two sides that form the right angle are the legs. The sides always satisfy the Pythagorean theorem, a² + b² = c².

What is the full Pythagorean theorem formula?

The Pythagorean theorem is a² + b² = c², where a and b are the legs and c is the hypotenuse. Solved for each side, it gives c = √(a² + b²) for the hypotenuse, a = √(c² − b²) for one leg, and b = √(c² − a²) for the other. For legs 3 and 4, c = √(9 + 16) = 5.

Is 4.5, 6 and 7.5 a right triangle?

Yes. 4.5² + 6² = 20.25 + 36 = 56.25, and 7.5² = 56.25, so the sides satisfy a² + b² = c². It is the 3-4-5 triangle scaled by 1.5. Any multiple of 3-4-5 is a right triangle, which is the 3-4-5 rule builders use to square corners, and the same goes for other Pythagorean triples such as 5-12-13 or 30-40-50.

Is this right triangle calculator free?

Yes. It is free, needs no sign-up, and runs entirely in your browser on phones, tablets, and desktops. Every result also gets a shareable link you can bookmark or send.

More questions? See the full right triangle FAQ.

Related calculators

Special right triangle calculator

Solve 30-60-90 and 45-45-90 triangles from a single known side.

Pythagorean theorem calculator

Find the missing side of a right triangle, or check if 3 sides form one.

30-60-90 triangle calculator

Solve a 30-60-90 triangle from a single known side.

Isosceles right triangle calculator

Solve a 45-45-90 isosceles right triangle from a single known side.

Right triangle formulas

Every right-triangle formula in one reference sheet.

Right triangle FAQ

Answers to common right-triangle questions.