The Pythagorean theorem is a way to find a missing side in a right triangle using the formula a2+b2=c2a^2+b^2=c^2a2+b2=c2, where ccc is the hypotenuse (the longest side, opposite the right angle) and aaa and bbb are the other two sides.

How to Do the Pythagorean Theorem

Quick Scoop guide to right triangles

1. What the theorem says

In any right triangle:

  • The hypotenuse is the side across from the 90° angle and is always the longest.
  • The other two sides are often called legs (or base and height).
  • The Pythagorean theorem says:

a2+b2=c2a^2+b^2=c^2a2+b2=c2

where ccc is the hypotenuse, and aaa and bbb are the legs.

A common way to say it in words: “The square of the hypotenuse is equal to the sum of the squares of the other two sides.”

2. Step‑by‑step: using the formula

Think of this as a simple 3‑step recipe.

Case A: Find the hypotenuse

Example: A right triangle has legs 3 and 4. Find the hypotenuse ccc.

  1. Write the formula

a2+b2=c2a^2+b^2=c^2a2+b2=c2

  1. Plug in the numbers
    • Let a=3a=3a=3, b=4b=4b=4.

32+42=c23^2+4^2=c^232+42=c2

9+16=c29+16=c^29+16=c2

25=c225=c^225=c2

  1. Take the square root

c=25=5c=\sqrt{25}=5c=25​=5

So the hypotenuse is 5.

Case B: Find a missing leg

Example: A right triangle has hypotenuse 13 and one leg 5. Find the other leg bbb.

  1. Start from the formula

a2+b2=c2a^2+b^2=c^2a2+b2=c2

  1. Plug in the known sides
    Let a=5a=5a=5, c=13c=13c=13.

52+b2=1325^2+b^2=13^252+b2=132

25+b2=16925+b^2=16925+b2=169

  1. Solve for the unknown
    Subtract 25 from both sides:

b2=169−25=144b^2=169-25=144b2=169−25=144

Then take the square root:

b=144=12b=\sqrt{144}=12b=144​=12

So the missing leg is 12.

3. Mini “story” to remember it

Imagine you draw a square on each side of a right triangle.

  • On each short side you get a smaller square.
  • On the hypotenuse you get a bigger square.
  • The area of the big square is exactly the same as the sum of the areas of the two smaller squares.

That picture idea is what the formula a2+b2=c2a^2+b^2=c^2a2+b2=c2 is really saying: “big square = small square + medium square.”

4. Common Pythagorean triples

These are “famous” side sets that automatically satisfy a2+b2=c2a^2+b^2=c^2a2+b2=c2.

Leg 1Leg 2Hypotenuse
345
51213
81517
72425
These are handy in homework or tests because you can often spot them **without** doing full calculations.

5. Quick checklist when you use it

Use this little mental checklist each time:

  1. Is the triangle a right triangle? (There must be a 90° angle.)
  1. Did you correctly identify the hypotenuse (across from the 90° angle)?
  1. Did you square the numbers before adding or subtracting?
  2. If you’re finding a leg, did you subtract from c2c^2c2, not add?
  3. Did you take the positive square root at the end? (Side lengths are positive.)

6. Where this shows up in real life

You’ll see the Pythagorean theorem in:

  • Distance problems on grids (like finding distance between two points on a map or coordinate plane).
  • Construction and carpentry, to make sure corners are exactly 90°.
  • Physics, coding, graphics: calculating straight‑line distances in 2D and 3D.

A simple example: If you walk 6 blocks east and 8 blocks north, your straight‑line distance from the start is 62+82=10\sqrt{6^2+8^2}=1062+82​=10.

7. Super‑short TL;DR

  • Formula: a2+b2=c2a^2+b^2=c^2a2+b2=c2 for right triangles only.
  • Use it to find a missing side when you know the other two.
  • Hypotenuse is always the longest side and goes on the c spot.

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Learn how to do the Pythagorean theorem step by step with simple examples, tips, and common triples. Understand how a2+b2=c2a^2+b^2=c^2a2+b2=c2 works and when to use it in right triangles.

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