All calculatorsRight Triangle Calculator
Solve a right triangle from two sides or from one side + one acute angle. Get all sides (a, b, c), angles (A, B), area, perimeter, and a clean mini triangle visual.
Background
In a right triangle, the hypotenuse c is opposite the 90° angle and is always the largest side. The Pythagorean theorem is a² + b² = c².
How this calculator works
- Two sides: uses a² + b² = c² to solve the missing side.
- Side + angle: uses sin(A)=a/c, cos(A)=b/c, tan(A)=a/b.
- Angles: computes A = asin(a/c) and B = 90° − A (or equivalents).
- Area: Area = (a·b)/2 • Perimeter: P = a + b + c
Tip: If you provide c and one leg, the other leg is √(c² − leg²), so c must be larger than that leg.
Formula & Equation Used
Pythagorean theorem: a² + b² = c²
Sine: sin(A) = a/c
Cosine: cos(A) = b/c
Tangent: tan(A) = a/b
Area: Area = (a·b)/2
Perimeter: P = a + b + c
Example Problems & Step-by-Step Solutions
Example 1 — Two sides (a=3, b=4)
Find c, A, and B.
- Use c = √(a² + b²) → c = √(3² + 4²) = √25 = 5
- Angle A = asin(a/c) = asin(3/5) ≈ 36.87°
- B = 90° − A ≈ 53.13°
- Area = (a·b)/2 = (3·4)/2 = 6
Example 2 — Two sides (c=13, b=12)
Find the missing leg a, plus A and B.
- Use a = √(c² − b²) → a = √(13² − 12²) = √(169 − 144) = √25 = 5
- Angle A = asin(a/c) = asin(5/13) ≈ 22.62°
- B = 90° − A ≈ 67.38°
- Area = (a·b)/2 = (5·12)/2 = 30
This is the classic 5–12–13 right triangle (the hypotenuse is 13).
Example 3 — Side + angle (A=30°, c=10)
Find a, b, and B.
- Use a = c·sin(A) → a = 10·sin(30°) = 10·0.5 = 5
- Use b = c·cos(A) → b = 10·cos(30°) = 10·(√3/2) ≈ 8.66
- B = 90° − A = 60°
- Area = (a·b)/2 ≈ (5·8.66)/2 ≈ 21.65
Frequently Asked Questions
Q: Which side is the hypotenuse?
The hypotenuse is the side opposite the 90° angle and it’s always the longest. In this calculator it’s c.
Q: Why does “c and a” sometimes fail?
Because you need c > a. Otherwise √(c² − a²) is not real.
Q: Do I use degrees or radians?
Either—choose it in the dropdown. The calculator displays angles in both degrees and radians.
