What is a right triangle?

A right triangle is a triangle that has one interior angle equal to 90 degrees. The side opposite the right angle is called the hypotenuse, and the other two sides are called legs. In a right triangle, the Pythagorean theorem holds: a^2 + b^2 = c^2.

Which inputs does this right triangle calculator accept?

You can enter any two of the following: the two legs a and b, a leg and the hypotenuse, the hypotenuse and an acute angle, or a leg and its opposite acute angle. The calculator checks that the combination is valid for a right triangle before solving.

Does this calculator use degrees or radians?

By default the calculator uses degrees for angles, but you can switch to radians from the angle unit selector. Internally, trigonometric functions are evaluated consistently with the chosen unit.

What quantities can I compute for the right triangle?

For a valid set of inputs the calculator returns all three sides, the two acute angles, area, perimeter, and the altitude to the hypotenuse. It also reports the Pythagorean check a^2 + b^2 and c^2 to highlight numerical consistency.

Right Triangle Calculator

Pythagoras + Trig

Solve any right triangle from two known values. Enter sides \(a, b, c\) (with \(c\) the hypotenuse) or acute angles \(\alpha\) and \(\beta\) and get all remaining sides, angles, area, perimeter and altitude, with a Pythagorean check.

Ideal for students, teachers, engineers and exam prep: consistent notation, unit label support, and formulas you can reuse in your own work.

Interactive right triangle solver

Angle unit

Controls how \(\alpha\) and \(\beta\) are interpreted and displayed.

Length unit (display label)

Optional label used only in the output (formulas are dimensionless).

Leg a

Side opposite angle \(\alpha\).

Leg b

Side opposite angle \(\beta\).

Hypotenuse c

Side opposite the right angle (must be the longest side).

Angle \(\alpha\) (at A)

Acute angle opposite side a. Must be between 0 and 90° (or 0 and \(\pi/2\) rad).

Angle \(\beta\) (at B)

Other acute angle. In a right triangle \(\alpha + \beta = 90^\circ\).

Input rule:

Provide exactly two of the five fields above. Supported pairs:

(a, b)
(a, c) or (b, c)
(c, \(\alpha\)) or (c, \(\beta\))
(a, \(\alpha\)) or (b, \(\beta\))

Solution

Quantity	Value

Right triangle basics

A right triangle has one angle equal to \(90^\circ\). We adopt the standard notation:

\(c\) – the hypotenuse, opposite the right angle.
\(a\) and \(b\) – the legs, adjacent to the right angle.
\(\alpha\) – the acute angle opposite side \(a\).
\(\beta\) – the acute angle opposite side \(b\).

The three key relationships are:

Pythagorean theorem: \[ a^2 + b^2 = c^2 \]
Angle sum: \[ \alpha + \beta = 90^\circ \quad (\text{or } \alpha + \beta = \pi/2 \text{ rad}) \]
Trigonometry (for angle \(\alpha\)): \[ \sin\alpha = \frac{a}{c}, \quad \cos\alpha = \frac{b}{c}, \quad \tan\alpha = \frac{a}{b} \]

Solving a right triangle from two known values

Given both legs \(a\) and \(b\)

If you know the two legs:

\[ c = \sqrt{a^2 + b^2} \] \[ \alpha = \arctan\left(\frac{a}{b}\right), \qquad \beta = \arctan\left(\frac{b}{a}\right) \]

The area is \(A = \frac{1}{2}ab\) and the perimeter is \(P = a + b + c\).

Given a leg and the hypotenuse

Suppose you know leg \(a\) and hypotenuse \(c\) (with \(c > a\)):

\[ b = \sqrt{c^2 - a^2} \] \[ \alpha = \arcsin\left(\frac{a}{c}\right), \qquad \beta = 90^\circ - \alpha \]

The case \((b, c)\) is analogous.

Given hypotenuse and angle

If you know \(c\) and an acute angle \(\alpha\):

\[ a = c \sin\alpha, \qquad b = c \cos\alpha \] \[ \beta = 90^\circ - \alpha \]

Given leg and its opposite angle

If you know leg \(a\) and angle \(\alpha\) opposite that leg:

\[ c = \frac{a}{\sin\alpha}, \qquad b = \frac{a}{\tan\alpha} \]

Again, \(\beta = 90^\circ - \alpha\). The case \((b, \beta)\) is symmetric.

Altitude to the hypotenuse

The altitude from the right angle to the hypotenuse (call it \(h_c\)) is related to the area in two ways:

\[ A = \frac{1}{2}ab = \frac{1}{2} c h_c \quad \Rightarrow \quad h_c = \frac{ab}{c} \]

This quantity appears in similarity proofs and in geometric constructions involving right triangles.