Pythagorean Theorem Calculator
Find any side of a right triangle with step-by-step solution.
One leg of the right triangle.
The other leg of the right triangle.
Use the same unit for every side.
Missing side c
5 units
Valid right triangle calculation
Formula used
c = √(a² + b²)
c = √(3² + 4²) = √25
Side a
3 units
Side b
4 units
Hypotenuse c
5 units
Exact form
5
a²
9
b²
16
c²
25
Pythagorean triple
Yes
Pythagorean Theorem Formulas
Pythagorean Theorem
a² + b² = c²
Find Hypotenuse
c = √(a² + b²)
Find Leg a
a = √(c² − b²)
Find Leg b
b = √(c² − a²)
Right Triangle Check
a² + b² = c²
Variable Explanations
a
One leg of the right triangle.
b
The other leg of the right triangle.
c
The hypotenuse.
Hypotenuse
Longest side, opposite the right angle.
Square
A number multiplied by itself.
Square root
The value that produces the squared number.
Right angle
A 90-degree angle.
Right-Triangle Diagram
The legs a and b meet at the right angle. The hypotenuse c is opposite the right angle and is always the longest side.
Worked Examples
Find hypotenuse from a = 3 and b = 4
Formula: c = √(a² + b²)
Substitution: c = √(3² + 4²) = √25
Answer: c = 5
3-4-5 is a Pythagorean triple.
Find hypotenuse from a = 5 and b = 12
Formula: c = √(a² + b²)
Substitution: c = √(25 + 144) = √169
Answer: c = 13
5-12-13 is another common triple.
Find missing leg from c = 13 and b = 5
Formula: a = √(c² − b²)
Substitution: a = √(13² − 5²) = √144
Answer: a = 12
Subtract the known leg squared from the hypotenuse squared.
Find missing leg from c = 10 and a = 6
Formula: b = √(c² − a²)
Substitution: b = √(10² − 6²) = √64
Answer: b = 8
The missing leg is 8.
Decimal result example
Formula: c = √(2² + 3²)
Substitution: c = √13
Answer: c ≈ 3.6055
Not every result is a whole number.
Invalid triangle example
Formula: a = √(c² − b²)
Substitution: a = √(5² − 8²)
Answer: Invalid
The hypotenuse cannot be shorter than a leg.
Right-triangle validation
Formula: a² + b² = c²
Substitution: 6² + 8² = 10²
Answer: 36 + 64 = 100
The sides form a right triangle.
Real-world diagonal
Formula: c = √(width² + height²)
Substitution: c = √(9² + 12²)
Answer: c = 15
Useful for finding diagonal distances.
Finding Hypotenuse vs Missing Leg
Finding the hypotenuse
Use addition: a² + b². The hypotenuse is unknown, so add both leg squares and take the square root.
Finding a missing leg
Use subtraction: c² − known leg². The hypotenuse must be known and must be the longest side.
Pythagorean Triples and Real-World Uses
Common Pythagorean triples include 3-4-5, 5-12-13, and 8-15-17. If units are used, all side lengths must use the same unit.
Common Pythagorean Theorem Mistakes
Understanding Your Results
Missing side length
The side needed to complete the right triangle.
Hypotenuse
The longest side opposite the right angle.
Missing leg
One of the two sides that form the right angle.
Squared values
Values used in the theorem before taking the square root.
Exact radical form
The symbolic result before decimal approximation.
Triangle status
Whether the side lengths satisfy the right-triangle relationship.
Frequently Asked Questions
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