The most important theorem in geometry — every right triangle obeys it, no exceptions.
The Mnemonic
One Equation, Every Right Triangle
The Pythagorean Theorem says: in any right triangle, if a and b are the two legs (the sides forming the right angle) and c is the hypotenuse (the side opposite the right angle, always the longest side), then a² + b² = c².
This only works for right triangles — it's not a general triangle rule. The hypotenuse is always isolated by itself on one side of the equation, and it's always the longest side, since it sits opposite the largest angle (90°).
Right triangle with legs a, b and hypotenuse c — the right angle sits opposite the hypotenuse
💡 Memory Trick
"The most important theorem in geometry — all right triangles obey it." a² + b² = c² — legs squared and added equal the hypotenuse squared. The hypotenuse is always alone on the c side, and it's always the longest side of the triangle, opposite the right angle.
Why It Works
Why Squaring the Sides Works
One classic way to see why this is true: draw a square directly on each side of a right triangle, using that side as the square's edge length. The area of the square on the hypotenuse turns out to be exactly equal to the combined area of the two squares built on the legs — literally, a² (area of the smaller square) plus b² (area of the medium square) equals c² (area of the largest square).
This is why the theorem is phrased in terms of squares rather than the side lengths directly — it's fundamentally a statement about area, not just length, even though it gets used constantly as a length-finding shortcut.
Using It In A Proof
Solving for Any Missing Side
The theorem can solve for any one of the three sides, as long as you know the other two — the algebra just changes slightly depending on which side is missing.
1
Identify the hypotenuse first
Always find the hypotenuse before plugging into the formula — it's the side opposite the right angle, and it must be isolated on its own side of the equation.
2
Solve for the hypotenuse (if that's missing)
Plug both legs directly into a² + b² = c², then take the square root of both sides at the end.
Legs 3 and 4: 3² + 4² = c² → 9 + 16 = 25 → c = 5
3
Solve for a leg (if that's missing instead)
Rearrange first: a² = c² − b². Subtracting happens BEFORE the square root, never after.
Hypotenuse 13, one leg 5: a² = 13² − 5² = 169 − 25 = 144 → a = 12
Full Worked Example
Finding the Hypotenuse of a Ladder Problem
Given: A right triangle has legs of length 9 and 12. Find: the hypotenuse c.
Legs of 9 and 12 — solve for the hypotenuse c
1
Confirm it's a right triangle
The problem states (or the diagram marks) a right angle between the two given legs — without this, the theorem doesn't apply at all.
2
Set up the equation
a² + b² = c² becomes 9² + 12² = c².
3
Simplify
81 + 144 = c², so c² = 225.
4
Take the square root
c = √225 = 15.
Notice 9-12-15 is just the 3-4-5 triangle scaled up by 3 — recognizing common scaled Pythagorean triples (3-4-5, 5-12-13, 8-15-17) can save time on future problems.
🎯 Quick Worked Example
A right triangle has a hypotenuse of 10 and one leg of 6. Find the other leg.
1
Set up the rearranged equation. Since the hypotenuse is known, solve for the missing leg: a² = c² − b², so a² = 10² − 6².
2
Simplify. a² = 100 − 36 = 64.
3
Take the square root. a = √64 = 8. (This is the 6-8-10 triangle — a scaled 3-4-5.)
📌 Exam Application
Recognizing the common Pythagorean triples — 3-4-5, 5-12-13, 8-15-17, and their multiples (6-8-10, 9-12-15, etc.) — lets you skip the arithmetic entirely on many problems. If two of the three numbers in a right-triangle problem match a scaled triple, the third is a fast mental calculation rather than a full square-root computation.
⚠️ Most Common Pythagorean Theorem Mistakes
Trap 1 — Forgetting to identify the hypotenuse correctly: The hypotenuse must always be isolated on its own side of the equation (as c²) — plugging it in as one of the two added terms instead produces a completely wrong answer.
Trap 2 — Taking the square root too early: When solving for a leg, you must complete the subtraction (c² − b²) BEFORE taking the square root — √c² − √b² is not a valid operation and gives a different (wrong) result than √(c² − b²).
✓ Quick Self-Test
1) State the Pythagorean Theorem. 2) Which side of a right triangle is always the longest? 3) A right triangle has legs 5 and 12 — find the hypotenuse. 4) A right triangle has a hypotenuse of 17 and one leg of 8 — find the other leg. 5) Does the Pythagorean Theorem work for triangles that aren't right triangles?