Right Triangle Calculator

Enter any two known values to calculate all sides, angles, area, perimeter, and more with step-by-step solutions.

Enter Any Two Values

Leave unknown fields empty. Fill in exactly two fields to solve the triangle.

Angle A is opposite leg a. Angle B is opposite leg b. Angle C = 90 degrees.

Results

Area
--
square units
Leg a--
Leg b--
Hypotenuse c--
Angle A--
Angle B--
Angle C90 degrees
Perimeter--
Altitude to Hypotenuse--
Inradius--
Circumradius--

Step-by-Step Solution

What Is a Right Triangle?

A right triangle is a triangle with one angle equal to 90 degrees. The side opposite the right angle is called the hypotenuse and is always the longest side. The other two sides are called legs. The Pythagorean theorem (a^2 + b^2 = c^2) relates the lengths of all three sides.

Key Formulas

Pythagorean Theorem

Relates the three sides of a right triangle.

c = sqrt(a^2 + b^2)

Area

Half the product of the two legs.

A = (1/2) x a x b

Altitude to Hypotenuse

The perpendicular distance from the right angle to the hypotenuse.

h = (a x b) / c

Inradius

Radius of the inscribed circle.

r = (a + b - c) / 2

Circumradius

Radius of the circumscribed circle.

R = c / 2

Trigonometric Relations

Relating sides and angles.

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

Solving a Right Triangle

To fully solve a right triangle, you need at least two pieces of information besides the right angle. This calculator accepts any combination of two known values from: leg a, leg b, hypotenuse c, or an acute angle. From these two values, all other measurements can be derived using the Pythagorean theorem and trigonometric functions.

Special Right Triangles

  • 3-4-5 triangle: The most common Pythagorean triple. Multiples like 6-8-10 and 9-12-15 are also valid.
  • 45-45-90 triangle: An isosceles right triangle where both legs are equal and the hypotenuse is leg x sqrt(2).
  • 30-60-90 triangle: The sides are in ratio 1 : sqrt(3) : 2.
  • 5-12-13 and 8-15-17: Other common Pythagorean triples.

Applications

Right triangles are fundamental in trigonometry, navigation, construction, and physics. They are used to calculate distances, heights of buildings, slopes, and vectors. Understanding right triangle relationships is essential for surveying, architecture, engineering, and many scientific disciplines.