Pythagorean Theorem Calculator
Enter any two sides of a right triangle and the calculator finds the missing third side using a² + b² = c². It also shows the triangle's area, all three angles, and the full substitution steps.
How It Works
The Pythagorean theorem states: for a right triangle with legs a and b and hypotenuse c,
a² + b² = c²
Three possible unknowns:
Solve for hypotenuse c (given a and b):
c = √(a² + b²)
Example: a=3, b=4 → c = √(9+16) = √25 = 5
Solve for leg a (given b and c):
a = √(c² − b²)
Example: b=4, c=5 → a = √(25−16) = √9 = 3
Solve for leg b (given a and c):
b = √(c² − a²)
Example: a=3, c=5 → b = √(25−9) = √16 = 4
Additional results computed:
• Triangle area = ½ × a × b
• Angle A (opposite side a) = arctan(a/b) in degrees
• Angle B (opposite side b) = arctan(b/a) in degrees
• Angle C = 90° (the right angle)
• Perimeter = a + b + c
Frequently Asked Questions
What is the hypotenuse?
The hypotenuse is the longest side of a right triangle and is always opposite the right angle (90°). In the formula a² + b² = c², c is the hypotenuse. It is always longer than either leg.
What are Pythagorean triples?
Pythagorean triples are sets of three whole numbers that satisfy a² + b² = c². The most famous is (3, 4, 5). Other common triples include (5, 12, 13), (8, 15, 17), and (7, 24, 25). Any multiple of a triple is also a triple — e.g., (6, 8, 10) is a multiple of (3, 4, 5).
Can the sides be decimal numbers?
Yes. The calculator works with any positive real numbers. Results are shown to 4 decimal places for precision.
Why must c be the longest side?
The hypotenuse is always opposite the right angle, which is the largest angle in the triangle (90°). In any triangle, the side opposite the largest angle is the longest. If you enter a "c" smaller than a or b, the calculation will produce an error (square root of a negative number).