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📐 Trigonometry & Geometry

Right Triangle Calculator

Solve any right-angled triangle. Enter any two sides, or one side and an acute angle, to find the hypotenuse, missing angles, area, and perimeter.

Enter Any 2 Values

Hypotenuse (c)
5.00
Angle α
36.87°
Area
6.00
Perimeter
12.00
a = 3 b = 4 c = 5

Step-by-Step Derivations

How to Use the Right Triangle Calculator

  1. Enter any two of the three sides — legs a and b, or either leg together with the hypotenuse c. Leave the fields you don't know blank.
  2. Alternatively, enter one side and the acute angle α (opposite side a, in degrees) to solve the triangle trigonometrically.
  3. Press Solve Triangle to compute the missing side, both acute angles α and β, the area, and the perimeter with step-by-step derivations.
  4. Check the SVG diagram: it redraws the triangle to scale from your computed legs a and b, so you can visually verify the proportions.

Formulas Used by This Calculator

Every right triangle obeys two complementary rule sets. The Pythagorean theorem links the three sides, while the SOH-CAH-TOA trigonometric ratios link the sides to the acute angles. Throughout this page the legs a and b meet at the right angle, the hypotenuse is c, and the angle α is opposite side a (between side b and the hypotenuse).

Pythagorean Theorem

a² + b² = c²

The square of the hypotenuse equals the sum of the squares of the two legs. Rearranged, it solves any missing side: c = √(a² + b²),  a = √(c² − b²),  b = √(c² − a²).

SOH-CAH-TOA Ratios

sin α = a / c
cos α = b / c
tan α = a / b

Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent — the key to finding a side from an angle (or vice versa with the inverse functions).

Area & Perimeter

Area = ½ × a × b   ·   Perimeter = a + b + c

The two legs are perpendicular, so either one can serve as the base and the other as the height — no separate height formula is needed.

Worked Example: the 3-4-5 Triangle

Suppose a right triangle has legs a = 3 and b = 4. Here is how the calculator solves it, step by step:

  1. Hypotenuse: c = √(a² + b²) = √(3² + 4²) = √(9 + 16) = √25 = 5.
  2. Angle α = arctan(a / b) = arctan(3 / 4) ≈ 36.87°.
  3. Angle β = 90° − α = 90° − 36.87° = 53.13° (the acute angles of a right triangle always sum to 90°).
  4. Area = ½ × a × b = ½ × 3 × 4 = 6.
  5. Perimeter = a + b + c = 3 + 4 + 5 = 12.

The result is the famous 3-4-5 Pythagorean triple — the smallest whole-number right triangle, used by builders for centuries to lay out perfect right angles.

Common Questions & Answers

Frequently Asked Questions

What is the Pythagorean Theorem?
The Pythagorean Theorem states that in any right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two legs (a and b): a² + b² = c².
How do trigonometric ratios apply to right triangles?
Using SOH-CAH-TOA: sin(θ) = Opposite / Hypotenuse, cos(θ) = Adjacent / Hypotenuse, and tan(θ) = Opposite / Adjacent.

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Reviewed & updated: October 2026 Maintained by the OnlineResult Study Tools Team Report an issue
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