Right Triangle Calculator
Calculate sides, angles, area, and perimeter of right triangles using the Pythagorean theorem and trigonometry. Perfect for students, engineers, and professionals.
Two Sides
Enter any two sides to calculate the third side and all angles.
The Pythagorean Theorem
The Pythagorean theorem is the foundation for solving right triangles. It states that in a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides:
a² + b² = c²
Where c is the hypotenuse (longest side)
Find Hypotenuse (c)
c = √(a² + b²)
Find Leg (a)
a = √(c² − b²)
Find Leg (b)
b = √(c² − a²)
SOHCAHTOA - Trigonometric Ratios
Remember the basic trigonometric ratios with the mnemonic SOHCAHTOA. These ratios relate angles to side lengths in right triangles:
SOH
sin(θ) = O / H
Sine = Opposite / Hypotenuse
CAH
cos(θ) = A / H
Cosine = Adjacent / Hypotenuse
TOA
tan(θ) = O / A
Tangent = Opposite / Adjacent
Special Right Triangles
Some right triangles have special angle measurements that make calculations easier. These are commonly used in geometry and trigonometry.
45-45-90 Triangle (Isosceles Right Triangle)
Side ratio: 1 : 1 : √2
Both legs are equal. The hypotenuse is √2 times a leg.
30-60-90 Triangle
Side ratio: 1 : √3 : 2
Short leg opposite 30°, long leg opposite 60°, hypotenuse opposite 90°.
Common Pythagorean Triples
Pythagorean triples are sets of three positive integers (a, b, c) that satisfy a² + b² = c². These are useful for quick calculations without a calculator.
| Triple | a² | b² | c² | Verification |
|---|---|---|---|---|
| 3-4-5 | 9 | 16 | 25 | 9 + 16 = 25 ✓ |
| 5-12-13 | 25 | 144 | 169 | 25 + 144 = 169 ✓ |
| 8-15-17 | 64 | 225 | 289 | 64 + 225 = 289 ✓ |
| 7-24-25 | 49 | 576 | 625 | 49 + 576 = 625 ✓ |
| 20-21-29 | 400 | 441 | 841 | 400 + 441 = 841 ✓ |
| 9-40-41 | 81 | 1600 | 1681 | 81 + 1600 = 1681 ✓ |
Related Calculators
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