Right Triangle Calculator

A right triangle (or right-angled triangle) is a three-sided polygon that contains exactly one interior angle measuring 90°. As the foundational shape of trigonometry, right triangles govern the definitions of sine, cosine, and tangent. Understanding the geometric relationships between the sides and angles of a right triangle is a core requirement in carpentry, navigation, physics, and vector analysis.

Our free Right Triangle Calculator is a multi-parameter solver. By inputting any two known values (such as a leg and an angle, or a leg and the hypotenuse), our tool calculates the remaining sides (legs a and b, hypotenuse c), unknown angles (α and β), altitude (h), area (A), and perimeter (P). You can enter angles in degrees or radians (using expressions like pi/3 or pi/4).

Anatomy of a Right Triangle

The components of a right triangle feature specific names and mathematical relationships:

• Hypotenuse (c): The longest side of the right triangle, located directly opposite the 90° right angle.
• Legs (a and b): The two shorter perpendicular sides that meet to form the 90° right angle.
• Angles (α and β): The two acute interior angles opposite legs a and b respectively. The sum of these two angles always equals exactly 90° (α + β = 90°).
• Altitude (h): The perpendicular line segment drawn from the 90° vertex to the hypotenuse. The altitude divides the original triangle into two smaller, similar right triangles that are also similar to the original.

Area and Perimeter Formulas

The perimeter and area of a right triangle are calculated using these standard formulas:

Perimeter (P) = a + b + c

Area (A) = 1/2 × a × b = 1/2 × c × h

Pythagorean Triangles

When all three sides of a right triangle are whole integers, it is called a Pythagorean triangle. The lengths form a Pythagorean triple. Common examples include:

• 3, 4, 5 (where 3² + 4² = 5²)
• 5, 12, 13 (where 5² + 12² = 13²)
• 8, 15, 17 (where 8² + 15² = 17²)

Special Right Triangles

Two types of special right triangles occur frequently in geometry because their side lengths follow fixed, predictable ratios:

1. The 30°-60°-90° Right Triangle

A right triangle containing acute interior angles of 30° and 60°. The side lengths opposite the 30°, 60°, and 90° angles always adhere to the fixed ratio:

1 : √3 : 2

If the side opposite the 60° angle (b) is 5, you can find the other sides easily:

• Leg a (opposite 30°): a = b / √3 = 5 / √3
• Hypotenuse c (opposite 90°): c = (b × 2) / √3 = 10 / √3

This ratio is highly useful for manually evaluating trigonometric functions for multiples of π/6 radians.

2. The 45°-45°-90° Right Triangle (Isosceles Right)

An isosceles right triangle contains two equal acute angles of 45° and two legs of equal length (a = b). The side lengths always follow the ratio:

1 : 1 : √2

If the hypotenuse (c) is 5, the leg lengths are calculated as:

a = b = c / √2 = 5 / √2

This ratio is essential for evaluating trigonometric functions for multiples of π/4 radians.

Solving Triangles with Trigonometric Functions

If you only know one side length and one acute angle, you can use the three primary trigonometric ratios to solve the remaining sides:

• Sine (sin): sin(α) = Opposite / Hypotenuse = a / c
• Cosine (cos): cos(α) = Adjacent / Hypotenuse = b / c
• Tangent (tan): tan(α) = Opposite / Adjacent = a / b

Determine coordinate side squares on our Pythagorean Theorem Calculator or analyze general oblique triangles using the main Triangle Calculator.

Frequently Asked Questions (FAQ)

What is the geometric meaning of the right triangle altitude?

The altitude (h) is the shortest line segment connecting the right-angle corner to the hypotenuse. Geometrically, it satisfies the relationship h = (a × b) / c. It splits the triangle into two sub-triangles that share the same angular proportions (similarity) as the outer triangle.

Why are 30°-60°-90° and 45°-45°-90° triangles called “special”?

They are called “special” because their side lengths follow exact algebraic ratios containing simple square roots (√3 and √2). This allows mathematicians and designers to calculate exact dimensions without converting to long, rounding decimal approximations.

How do you convert degrees to radians in a right triangle?

To convert an angle from degrees to radians, multiply the degree measurement by π / 180. For example, a 30° angle becomes 30 × (π / 180) = π / 6 radians, and a 45° angle becomes 45 × (π / 180) = π / 4 radians.

What is the hypotenuse of a right triangle with legs of 6 and 8?

Using the Pythagorean equation (a² + b² = c²):

c = √(6² + 8²) = √(36 + 64) = √100 = 10

This is a scaled version of the 3-4-5 Pythagorean triple, multiplied by a factor of 2.