Right Triangle Calculator

Why Use This Tool?

This right triangle calculator helps you find all unknown sides and angles of a right triangle when you know just two values. Whether you're a student, engineer, architect, or DIY enthusiast, this tool saves time and ensures accurate calculations for any right triangle problem.

How It Works

A right triangle has one 90-degree angle (angle C) and two acute angles (A and B) that sum to 90 degrees. Using the Pythagorean theorem (a² + b² = c²) and trigonometric functions, this calculator can determine all unknown values from any two known measurements.

b a c (hypotenuse) C (90°) A B

Formulas Used:

  • Pythagorean Theorem: a² + b² = c²
  • Sine: sin(A) = a/c, sin(B) = b/c
  • Cosine: cos(A) = b/c, cos(B) = a/c
  • Tangent: tan(A) = a/b, tan(B) = b/a
  • Angle Sum: A + B = 90°

Step-by-Step Instructions

  1. Look at the triangle diagram above to understand which side or angle corresponds to each input
  2. Enter any two known values (e.g., two sides, or one side and one angle)
  3. Click "Calculate" to find all unknown values
  4. The calculator will display all sides, angles, area, and perimeter
  5. Use "Clear All" to reset and start a new calculation

Calculator

Enter any two values to calculate the rest:

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Worked example: the 3-4-5 triangle

Enter side a = 3 and side b = 4, leaving everything else blank. With two sides known, the calculator applies the Pythagorean theorem directly:

c = √(a² + b²) = √(3² + 4²) = √(9 + 16) = √25 = 5

The angles come from inverse trig functions:

Angle A = arctan(a/b) = arctan(3/4) ≈ 36.87°
Angle B = 90° − 36.87° = 53.13°

Area and perimeter follow directly: area = (a×b)/2 = (3×4)/2 = 6, and perimeter = a+b+c = 3+4+5 = 12. The 3-4-5 triangle is the smallest right triangle with whole-number sides, which is why it's the standard reference example in geometry.

How the calculator picks which formula to use

The specific formula applied depends entirely on which two values you provide. Two sides known uses the Pythagorean theorem (as above). One side plus one angle uses sine, cosine, or tangent depending on which side and angle you supplied — for example, side a and angle A use c = a/sin(A) and b = a/tan(A). Three sides given (no angles) skips straight to verifying the Pythagorean identity and computing angles from there.

Validation the tool performs

Beyond requiring at least two inputs, the calculator checks that the hypotenuse is genuinely the longest side (rejecting, say, c=3 paired with a=5, which can't form a valid right triangle), and that any two supplied angles actually sum to 90° — if they don't, it flags the input as inconsistent rather than silently computing a wrong triangle.

What two angles alone cannot give you

Knowing angle A and angle B (which will always sum to 90° in a right triangle) tells you the triangle's shape but not its size — infinitely many similar triangles share those same two angles. The calculator requires at least one side length to pin down actual measurements.