Pythagorean Theorem Guide: Formula and Examples
Use a^2 + b^2 = c^2 to solve right-triangle side lengths.
Quick Answer
c = √(a² + b²); a = √(c² − b²), for a right triangleWhen to Use This
Use the theorem only when one angle is 90°. The hypotenuse c lies opposite that angle and is the longest side. Finding a length is different from finding area: intermediate squares use squared units, but the final square root returns a length.
Step-by-Step Method
- Mark the right angle and identify the hypotenuse.
- To find the hypotenuse, add the squares of the two legs.
- To find a leg, subtract the square of the known leg from the square of the hypotenuse.
- Take the positive square root and check the resulting side lengths.
Calculator Tips and Limits
- Open the main calculator, choose Scientific and enter sqrt(3^2 + 4^2) using the root and power controls.
- For a missing leg use sqrt(13^2 - 5^2). The linked calculator evaluates arithmetic; you identify and arrange the triangle sides.
Worked Examples
Find the hypotenuse
Legs of 3 cm and 4 cm give c² = 3² + 4² = 9 + 16 = 25 cm². Taking the positive square root gives c = 5 cm. Check: 5 exceeds both legs and 3² + 4² = 5².
Find a missing leg
With hypotenuse 13 m and one leg 5 m, the other leg satisfies a² = 13² - 5² = 169 - 25 = 144. Thus a = 12 m. Adding rather than subtracting would solve a different problem.
Common Mistakes
Using area units for a side
The answer 5 cm is a length. 25 cm² is its square, not the requested side length.
Choosing an impossible hypotenuse
If a proposed hypotenuse is shorter than a leg, the squared difference becomes negative. Recheck the labels and measurements.
FAQ
Why use the positive square root?
The algebraic equation has positive and negative roots, but a physical side length is positive.
Does the theorem apply to a non-right triangle?
Not in this form. You need further information and an appropriate method, such as the cosine rule.
References and corrections
OpenStax: Prealgebra 2e — further reading. The worked examples on this page are CalcSolver explanations. Read our calculation and correction method, or report an error with the input and expected result.