Right Triangle Calculator

Right triangle calculator at a glance#

A right triangle calculator solves for the sides, angles and area of a triangle with one 90-degree angle. The two short sides are the legs, and the longest side, opposite the right angle, is the hypotenuse. It uses the Pythagorean theorem for the sides and trigonometric ratios for the angles.

For the sides, a² + b² = c². For the angles, each acute angle comes from a ratio of two sides: sine = opposite / hypotenuse, cosine = adjacent / hypotenuse, tangent = opposite / adjacent. The area is half the product of the two legs: area = ½ × leg1 × leg2.

Worked example with legs a = 3 and b = 4. The hypotenuse is c = the square root of (3² + 4²) = the square root of 25 = 5. The area is ½ × 3 × 4 = 6 square units. For the angle opposite the side of length 3, sine = 3 / 5 = 0.6, which gives about 36.87 degrees; the angle opposite the side of length 4 is about 53.13 degrees. The two acute angles add up to 90 degrees, and all three angles add up to 180 degrees.

Enter any two known values, two sides or one side and one angle, in the calculator above for the exact remaining sides, angles, area and perimeter. Angle results are rounded to a set number of decimals, so a displayed value may differ slightly from a by-hand calculation.

What you can solve#

A right triangle has one 90-degree angle, two legs, and a hypotenuse opposite the right angle. Give the calculator any two values, two sides or one side and one acute angle, and it returns the rest: the missing sides, both acute angles, the area and the perimeter.

Solving the angles with trigonometry#

Once you know two sides, each acute angle comes from a trig ratio: sine = opposite / hypotenuse, cosine = adjacent / hypotenuse, tangent = opposite / adjacent. Take the inverse function to get the angle. For a 3-4-5 triangle, the angle opposite the side of length 3 has sine = 3/5 = 0.6, so the angle is arcsin(0.6) ≈ 36.87 degrees. The other acute angle is 90 − 36.87 ≈ 53.13 degrees, and the two always sum to 90.

Area and perimeter#

The area of a right triangle is half the product of its two legs: area = ½ × leg1 × leg2. For legs 3 and 4 that is ½ × 3 × 4 = 6 square units. The perimeter is the sum of all three sides: 3 + 4 + 5 = 12 units.

Special right triangles#

Two right triangles have fixed side ratios worth memorizing. In a 45-45-90 triangle the legs are equal and the hypotenuse is √2 times a leg, so legs of 1 give a hypotenuse of about 1.414. In a 30-60-90 triangle the sides are in the ratio 1 : √3 : 2, so the side opposite the 30-degree angle is half the hypotenuse and the side opposite 60 degrees is √3 times the shortest side.

Frequently asked questions#

What is a right triangle?#

A triangle with one 90-degree angle. The side opposite the right angle is the hypotenuse and is always the longest side; the other two sides are the legs.

What two values does the calculator need?#

Any two, as long as one is a side: two sides, or one side and one acute angle. From those it finds the remaining sides, both angles, the area and the perimeter.

How do I find the angles of a right triangle?#

Use a trig ratio of two sides, then its inverse. For sine = opposite / hypotenuse = 0.6, the angle is arcsin(0.6) ≈ 36.87 degrees. The three angles always add to 180, and the two acute ones add to 90.

How do I find the area?#

Multiply the two legs and halve the result: area = ½ × leg1 × leg2. The legs act as the base and height because they meet at the right angle. Legs of 6 and 8 give an area of 24 square units.

How is this different from the Pythagorean theorem calculator?#

The Pythagorean theorem only relates the three side lengths (a² + b² = c²). This calculator goes further, using trigonometry to solve angles and returning area and perimeter, so you can start from a side and an angle, not just two sides.