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Right Triangle Calculator

Solve right triangles from any two known values. Computes all sides, angles, area, and perimeter using the Pythagorean theorem and basic trigonometry.

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Classic 3-4-5 Right Triangle

The simplest Pythagorean triple — legs 3 and 4, hypotenuse 5.

Key values: a = 3 · b = 4 · c = 5

Roof Rafter Calculation

A roof with a horizontal run of 5 m and vertical rise of 2.5 m — what is the rafter length?

Key values: run = 5 m · rise = 2.5 m · rafter ≈ 5.59 m

Solve Any Triangle

Select your known values (SSS, SAS, ASA, AAS, or SSA) and enter measurements to solve the triangle and compute area, perimeter, inradius, and more.

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Equilateral Triangle

Equilateral Triangle Calculator | Side, Area & Height

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Right Triangle Properties

A right triangle has one angle of exactly 90°. The side opposite the right angle is called the hypotenuse (always the longest side), and the other two sides are called legs.

The Pythagorean theorem governs right triangles:

a^2 + b^2 = c^2

where $c$ is the hypotenuse and $a, b$ are the legs. This calculator pre-sets the right angle at C = 90°, so you only need to provide two sides or one side and one acute angle.

Trigonometric Ratios (SOH-CAH-TOA)

The six trigonometric functions are defined using the sides of a right triangle relative to an acute angle $\theta$ :

SOH

\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}

CAH

\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}

TOA

\tan\theta = \frac{\text{opposite}}{\text{adjacent}}

The reciprocal ratios are cosecant ( $\csc = 1/\sin$ ), secant ( $\sec = 1/\cos$ ), and cotangent ( $\cot = 1/\tan$ ).

Special Right Triangles

Two right triangles have exact side ratios that appear constantly in geometry, trigonometry, and standardized tests:

45-45-90 Triangle

1 : 1 : \sqrt{2}

The isosceles right triangle. Both legs are equal, and the hypotenuse is $\sqrt{2}$ times a leg. Formed by cutting a square along its diagonal.

30-60-90 Triangle

1 : \sqrt{3} : 2

Formed by bisecting an equilateral triangle. The short leg (opposite 30°) is half the hypotenuse. The long leg (opposite 60°) is $\sqrt{3}$ times the short leg.

Solving Right Triangles

Given any two measurements (besides the right angle), you can find everything else:

Two legs known: Use Pythagorean theorem for hypotenuse, inverse trig for angles: $\theta = \arctan(a/b)$
Leg + hypotenuse: Second leg via $b = \sqrt{c^2 - a^2}$ , angles via inverse trig
One side + one acute angle: Use trig ratios to find the other sides, then $\text{third angle} = 90° - \theta$

The area of a right triangle is particularly simple:

A = \frac{1}{2} \times a \times b

where $a$ and $b$ are the legs (they form the base and height automatically).

Worked Examples

Ladder Against a Wall

A 13-foot ladder leans against a wall, with its base 5 feet from the wall. How high does it reach?

h = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12 \text{ ft}

Angle with the ground: $\theta = \arccos(5/13) \approx 67.4°$ .

Surveying a River Width

Standing 50 m from a river bank, you sight a tree on the far side at a 32° angle. The river width is:

w = 50 \times \tan(32°) \approx 50 \times 0.6249 \approx 31.2 \text{ m}

Frequently Asked Questions

What is the Pythagorean theorem?

The Pythagorean theorem states that in a right triangle, $a^2 + b^2 = c^2$ , where $c$ is the hypotenuse (the side opposite the right angle) and $a$ and $b$ are the two legs. It is the fundamental relationship governing all right triangles.

What is SOH-CAH-TOA?

SOH-CAH-TOA is a mnemonic for the three basic trigonometric ratios: $\sin\theta = \text{opposite}/\text{hypotenuse}$ (SOH), $\cos\theta = \text{adjacent}/\text{hypotenuse}$ (CAH), and $\tan\theta = \text{opposite}/\text{adjacent}$ (TOA). These ratios relate the sides of a right triangle to its acute angles.

What are the 30-60-90 and 45-45-90 special triangles?

These are right triangles with exact side ratios. A 45-45-90 triangle has sides in the ratio $1 : 1 : \sqrt{2}$ (formed by cutting a square diagonally). A 30-60-90 triangle has sides in the ratio $1 : \sqrt{3} : 2$ (formed by bisecting an equilateral triangle). These ratios are used extensively in geometry and trigonometry.

How do I find a missing side of a right triangle?

If you know two legs, find the hypotenuse with $c = \sqrt{a^2 + b^2}$ . If you know the hypotenuse and one leg, find the other leg with $b = \sqrt{c^2 - a^2}$ . If you know one side and one acute angle, use trigonometric ratios (sine, cosine, or tangent) to find the missing sides.

What are Pythagorean triples?

Pythagorean triples are sets of three positive integers that satisfy $a^2 + b^2 = c^2$ . Common examples include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). Any multiple of a triple is also a triple, so (6, 8, 10) works because it is twice (3, 4, 5).

Right Triangle Calculator

Solve right triangles from any two known values. Computes all sides, angles, area, and perimeter using the Pythagorean theorem and basic trigonometry.

Solve Mode

Known Values

Try an Example

Classic 3-4-5 Right Triangle

Roof Rafter Calculation

Solve Any Triangle

Related triangle-type Variants

Equilateral Triangle

More Math Calculators

Solve Mode

Known Values

Try an Example

Classic 3-4-5 Right Triangle

Roof Rafter Calculation

Solve Any Triangle

Right Triangle Properties

Trigonometric Ratios (SOH-CAH-TOA)

Special Right Triangles

45-45-90 Triangle

30-60-90 Triangle

Solving Right Triangles

Worked Examples

Ladder Against a Wall

Surveying a River Width

Frequently Asked Questions

What is the Pythagorean theorem?

What is SOH-CAH-TOA?

What are the 30-60-90 and 45-45-90 special triangles?

How do I find a missing side of a right triangle?

What are Pythagorean triples?

Right Triangle Calculator

Solve right triangles from any two known values. Computes all sides, angles, area, and perimeter using the Pythagorean theorem and basic trigonometry.

Solve Mode

Known Values

Try an Example

Classic 3-4-5 Right Triangle

Roof Rafter Calculation

Solve Any Triangle

Related triangle-type Variants

Equilateral Triangle

More Math Calculators

Calculator Search

Solve Mode

Known Values

Try an Example

Classic 3-4-5 Right Triangle

Roof Rafter Calculation

Solve Any Triangle

Right Triangle Properties

Trigonometric Ratios (SOH-CAH-TOA)

Special Right Triangles

45-45-90 Triangle

30-60-90 Triangle

Solving Right Triangles

Worked Examples

Ladder Against a Wall

Surveying a River Width

Frequently Asked Questions

What is the Pythagorean theorem?

What is SOH-CAH-TOA?

What are the 30-60-90 and 45-45-90 special triangles?

How do I find a missing side of a right triangle?

What are Pythagorean triples?