Geometry & Triangles

Pythagorean Theorem: How to Find a Missing Side

The Pythagorean theorem applies only to right triangles, with c representing the hypotenuse.

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Quick answer

The Pythagorean theorem applies only to right triangles, with c representing the hypotenuse.

Formula or method

a² + b² = c²

Keep units, percentages and time periods consistent. The formula is useful as a transparent check on the calculator result and helps explain why changing one input changes the answer.

Step-by-step

  1. Identify the hypotenuse.
  2. Square known side lengths.
  3. Add squares to find c or subtract to find a leg.
  4. Take the positive square root.

Worked example

For legs 3 and 4, c=5. With hypotenuse 13 and one leg 5, the other leg is 12.

How to interpret the result

The formula is specific to right triangles; other triangles require different methods.

When the answer is used for a purchase, loan, payroll decision, construction order, school grade, health estimate or other real-world choice, verify the assumptions that matter in that context. Accurate arithmetic still depends on accurate inputs.

Common mistakes

  • Misidentifying the hypotenuse.
  • Adding when solving for a leg.
  • Using the formula on a non-right triangle.

Frequently asked questions

Which side is c?

The side opposite the 90° angle.

Can it find angles?

Not by itself.

Why use positive square root?

Lengths are positive.

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