CALTRX

Right Triangle Calculator

Enter any two sides of a right triangle to calculate the third side, angles, area, and perimeter.

c = √(a² + b²)
a= First leg
b= Second leg
c= Hypotenuse (longest side)

Tips & Notes

  • ✓The hypotenuse is always the longest side of a right triangle.
  • ✓The two acute angles always sum to exactly 90°.
  • ✓The classic 3-4-5 triangle is the simplest Pythagorean triple.
  • ✓Leave hypotenuse as 0 to compute it from the two legs automatically.

Common Mistakes

  • ✗Entering the hypotenuse as a leg, producing an invalid triangle.
  • ✗Forgetting that a leg must be shorter than the hypotenuse.
  • ✗Using radians when degrees are expected for angle results.
  • ✗Applying the Pythagorean theorem as a² + b² = c instead of c².

Right Triangle Calculator Overview

What This Calculator Does

The Right Triangle Calculator takes two known sides of a right triangle and computes all remaining properties: the missing side via the Pythagorean theorem, both acute angles via inverse trigonometry, area, and perimeter. Every step is shown with the actual numeric values from your inputs.

The Pythagorean Theorem

For a right triangle with legs a and b and hypotenuse c: a² + b² = c². This relationship, known for over 2,500 years, connects the three sides and allows computing any one from the other two. Given two legs, the hypotenuse is √(a² + b²). Given the hypotenuse and one leg, the other leg is √(c² - a²).

Trigonometric Ratios

The acute angles are computed using inverse tangent: angle A = arctan(a/b), angle B = arctan(b/a). These two angles always sum to exactly 90°. The sine, cosine, and tangent of each angle relate the sides: sin(A) = opposite/hypotenuse, cos(A) = adjacent/hypotenuse, tan(A) = opposite/adjacent.

Practical Applications

Construction workers use right triangles to verify that corners are square using the 3-4-5 method. Architects calculate roof pitch as the ratio of rise to run. Surveyors determine distances to inaccessible points using triangulation. Physicists resolve force vectors into horizontal and vertical components. Navigation systems compute distances using right-triangle relationships between latitude and longitude differences.

Special Right Triangles

Two right triangles appear frequently enough to memorize: the 45-45-90 triangle (legs equal, hypotenuse = leg×√2) and the 30-60-90 triangle (sides in ratio 1:√3:2). Pythagorean triples are integer-sided right triangles: 3-4-5, 5-12-13, 8-15-17, and 7-24-25 are the most common.

Frequently Asked Questions

Enter the known leg and hypotenuse. Leave the other leg as 0 or enter any value — the calculator will derive the missing side from c² - a².

A set of three positive integers that satisfy a² + b² = c². Common examples: 3-4-5, 5-12-13, 8-15-17, 7-24-25.

Check that a² + b² = c² and that angle A + angle B = 90°. Both conditions must hold for a valid right triangle.

The Right Triangle Calculator uses standard validated formulas and provides results accurate to multiple decimal places. Review the step-by-step explanation to verify each calculation.