Right Triangle Calculator

Use this right triangle calculator to solve all missing sides and acute angles from two known sides or from one side plus an acute angle. It also calculates area, perimeter, altitude to the hypotenuse, inradius, circumradius, and common trigonometric ratios. A separate special-triangle mode handles exact 30-60-90 and 45-45-90 relationships.

How a right angle triangle calculator solves the triangle

A right triangle has one angle equal to 90°. The side opposite that right angle is the hypotenuse, and it is always the longest side. If the two legs are known, the missing hypotenuse follows from the Pythagorean theorem:

a² + b² = c²

where a and b are the legs and c is the hypotenuse. If one leg and the hypotenuse are known, rearrange the same formula, for example b = √(c² − a²).

Right triangle angle calculator

When the side lengths are known, the acute angles can be found with inverse trigonometric functions. In this calculator, angle A is opposite side a and adjacent to side b:

sin(A) = a/c
cos(A) = b/c
tan(A) = a/b

The other acute angle is B = 90° − A. OpenStax describes the same opposite/adjacent/hypotenuse relationships for right-triangle trigonometry.

Right triangle side calculator

If you know one side and one acute angle, choose the side that is known and enter angle A. For example, with hypotenuse c = 10 and A = 30°, the calculator gives a = 5 and b ≈ 8.6603. If the known value is the opposite leg instead, tangent or sine can be rearranged to recover the remaining sides.

Two angles alone determine only the shape of a triangle, not its size, so at least one side length is always required.

Special right triangle calculator

30-60-90 triangle

The side ratio opposite 30°, 60°, and 90° is:

1 : √3 : 2

If the short leg is 5, the long leg is 5√3 ≈ 8.6603 and the hypotenuse is 10.

45-45-90 triangle

The two legs are equal and the side ratio is:

1 : 1 : √2

If either leg is 5, the hypotenuse is 5√2 ≈ 7.0711.

Right triangle area calculator

The two legs are perpendicular, so one can be used as the base and the other as the height:

Area = a × b ÷ 2

That is the same formula people are looking for with searches such as area of a right triangle calculator. A 3-4-5 triangle has area 3 × 4 ÷ 2 = 6 square units.

Additional geometry results

The calculator also reports several values that are useful in geometry and construction problems:

• Perimeter: a + b + c
• Altitude to the hypotenuse: ab/c
• Inradius: (a + b − c)/2
• Circumradius: c/2

These follow directly from standard right-triangle geometry after the three side lengths are known.

Worked 3-4-5 example

Enter leg a = 3 and leg b = 4. The calculator finds c = 5. Angle A ≈ 36.8699°, angle B ≈ 53.1301°, area = 6, and perimeter = 12. It also reports the altitude to the hypotenuse as 2.4 and the inradius as 1.

Frequently asked questions

What inputs does the right triangle calculator need?

Two independent measurements are enough as long as at least one is a side: two sides, or one side and one acute angle.

Can I use this as a triangle calculator right-side solver?

Yes. The calculator assumes the third angle is exactly 90° and solves the remaining sides and angles under that right-triangle condition.

How do I know which side is the hypotenuse?

The hypotenuse is opposite the 90° angle and is always the longest side. In this calculator it is labeled c.

Does the calculator work in radians?

Yes for the side-plus-angle input mode. Select radians before entering angle A. Results are still displayed in degrees and radians for easy checking.

Methodology and sources

The solver uses the Pythagorean theorem, standard sine/cosine/tangent right-triangle relationships, and the exact 30-60-90 and 45-45-90 side ratios. These relationships are cross-checked against OpenStax right-triangle trigonometry material. The calculator performs geometry arithmetic only; it does not replace structural design or surveying requirements where professional measurements are required.