Calculat.dev LogoCalculat.dev - All Calculators, One Place.devAll Calculators, One Place.
Math CalculatorsPopular Tool Runs Locally

Right-Angled Triangle Calculator

Solve right triangles with hypotenuse, legs, area, perimeter, altitude, and acute angles using trigonometry and the Pythagorean theorem.

Right Triangle Solution
Hypotenuse (c)
5
Area
6
Angle α
36.87°
Angle β
53.13°
Calculation Steps:
Given legs a = 3, b = 4
Pythagorean theorem: c = √(a² + b²) = √(9 + 16) = √(25) = 5
Area: A = ½ × a × b = 0.5 × 3 × 4 = 6
Perimeter: P = a + b + c = 12
Altitude to hypotenuse: h = (a × b) / c = 2.4
Acute angles: α = arctan(a/b) = 36.87°, β = 90° - α = 53.13°
🎓

Did this calculation save you time?

Student Project

Hi! I'm a student developer building Calculat in my spare time. I was tired of searching for basic math tools and having to click through 10 spammy popups, loan ads, and cookie trackers.

I keep this website 100% free, private, and ad-free. If this helped you with your homework, project, or finances today, bookmarking this page or telling a friend helps me keep building more free tools!

Press ⌘ + D to Bookmark
Suggest What I Build Next

What Is a Right-Angled Triangle Calculator?

A right-angled triangle is a triangle in which one interior angle is exactly 90 degrees (a right angle). The side opposite the right angle is called the hypotenuse, and the other two sides are called legs or catheti.

How to Use This Calculator

  1. Enter leg a and leg b.
  2. Review the calculated hypotenuse (c), area, perimeter, and altitude.
  3. View the acute angles α and β in degrees.

Right Triangle Geometric Formulas

c = \sqrt{a^2 + b^2}, \quad A = \frac{1}{2}ab, \quad h_c = \frac{ab}{c}, \quad \alpha = \arctan\left(\frac{a}{b}\right)

Hypotenuse c is found using Pythagorean theorem. Area is half base times height. Altitude to hypotenuse is the product of legs divided by hypotenuse.

Worked Example

Scenario: Find hypotenuse and area for legs a = 6 and b = 8

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

Area: A = ½ × 6 × 8 = 24

Perimeter: P = 6 + 8 + 10 = 24

Angle α = arctan(6/8) ≈ 36.87°, Angle β = 90° - 36.87° = 53.13°

Result: Hypotenuse = 10, Area = 24, Angles: 36.87° and 53.13°

Tips & Key Notes

  • The two non-right angles in a right triangle are always complementary (they sum to 90°).
  • Common Pythagorean triples include 3-4-5, 5-12-13, 8-15-17, and 7-24-25.
  • The hypotenuse is always the longest side of any right triangle.

Frequently Asked Questions

What is the hypotenuse?

The hypotenuse is the side opposite the 90-degree right angle and is always the longest side of a right-angled triangle.

How do you find acute angles in a right triangle?

Use inverse trigonometric functions: angle α = arctan(opposite / adjacent) or arcsin(opposite / hypotenuse).

What is the altitude to the hypotenuse?

It is the perpendicular distance from the 90° vertex to the hypotenuse: h = (a × b) / c.

What are special right triangles?

Special right triangles include 45°-45°-90° (isosceles right triangle, ratio 1:1:√2) and 30°-60°-90° (ratio 1:√3:2).

Related Calculators

Explore similar tools
PopularMath

Pythagorean Theorem Calculator

Solve a² + b² = c² for missing hypotenuse or leg with step-by-step radical simplification and exact decimal solutions.

Calculate
PopularMath

Triangle Calculator

Calculate triangle area using Heron’s formula or base and height, perimeter, semi-perimeter, inradius, and classification.

Calculate
PopularMath

Sine Calculator (sin)

Calculate the sine of any angle in degrees or radians. Shows unit circle coordinates, exact values for special angles, and step-by-step steps.

Calculate