Pythagorean Theorem: Formula, Examples, and How to Use It

The Pythagorean theorem is one of the most useful formulas in all of math. It works on every standardized test from middle school through the GRE — and it powers everything from GPS to architecture to video-game physics. Best of all, the math itself is one line.

This guide gives you the formula, the proof intuition, every variation, the must-know triples, and the most common test-day applications.

The Formula

For any right triangle with legs \(a\) and \(b\) and hypotenuse \(c\):

\[a^2 + b^2 = c^2\]

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The hypotenuse is always the side opposite the right angle — the longest side. The other two sides are the legs.

The Idea, in One Picture

Imagine three squares attached to the three sides of a right triangle. The area of the square on the hypotenuse equals the sum of the areas of the squares on the two legs.

That is the theorem, geometrically. Algebraically: \(a^2 + b^2 = c^2\).

Finding the Hypotenuse

When you know the two legs:

Example 1

Legs are 3 and 4. Find the hypotenuse.

\(c^2 = 3^2 + 4^2 = 9 + 16 = 25\).

\(c = 5\).

Example 2

Legs are 5 and 12. Find the hypotenuse.

\(c^2 = 25 + 144 = 169\).

\(c = 13\).

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Finding a Leg

When you know one leg and the hypotenuse, rearrange:

\[a^2 = c^2 – b^2\]

Example 3

Hypotenuse is 10, one leg is 6. Find the other leg.

\(a^2 = 10^2 – 6^2 = 100 – 36 = 64\).

\(a = 8\).

Example 4

Hypotenuse is 17, one leg is 15.

\(a^2 = 289 – 225 = 64\).

\(a = 8\).

The Pythagorean Triples to Memorize

A Pythagorean triple is three whole numbers that satisfy \(a^2 + b^2 = c^2\). Memorize the common ones — they show up over and over on tests:

Every multiple of these is also a triple. So 6-8-10 (double of 3-4-5), 9-12-15 (triple), and 30-40-50 are all Pythagorean triples.

When you see a right triangle with two sides that look like the start of a triple, the third side is almost always the triple’s third number.

Special Right Triangles

These two right triangles come up so often that you should memorize the side ratios:

45°-45°-90° (Isosceles Right Triangle)

Legs are equal. The hypotenuse is leg × \(\sqrt{2}\).

Ratio: \(1 : 1 : \sqrt{2}\).

If a leg is 5, the hypotenuse is \(5\sqrt{2}\).

30°-60°-90°

Sides are in ratio \(1 : \sqrt{3} : 2\).

If the short leg is 4, the long leg is \(4\sqrt{3}\) and the hypotenuse is 8.

These ratios save time on the SAT, ACT, and GRE.

How to Check if a Triangle Is a Right Triangle

Given three side lengths, plug into \(a^2 + b^2 = c^2\) (using the largest as \(c\)).

• If the equation is true → right triangle.
• If \(a^2 + b^2 > c^2\) → acute triangle.
• If \(a^2 + b^2 < c^2\) → obtuse triangle.

Example 5

Is a triangle with sides 6, 8, 10 a right triangle?

\(6^2 + 8^2 = 36 + 64 = 100\).

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\(10^2 = 100\).

Yes — it is a right triangle.

Example 6

Is a triangle with sides 7, 8, 12 a right triangle?

\(7^2 + 8^2 = 49 + 64 = 113\).

\(12^2 = 144\).

$113 < 144$ → obtuse triangle, not a right triangle.

Real-World Applications

Distance between two points

The Pythagorean theorem is the distance formula. The distance between \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}\]

The horizontal change and vertical change are the legs. The straight-line distance is the hypotenuse.

Example 7

Distance from $(1, 2)$ to $(4, 6)$:

\(d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\).

Ladder problems

A ladder leans against a wall. The ladder is the hypotenuse; the wall and the ground are the legs.

Example 8: A 13-foot ladder rests against a wall, 5 feet from the base. How high up the wall does it reach?

\(5^2 + h^2 = 13^2\) → \(h^2 = 144\) → \(h = 12\) feet.

Diagonal of a rectangle

The diagonal of a rectangle with sides \(l\) and \(w\) is \(\sqrt{l^2 + w^2}\). Useful for figuring out whether a TV fits a wall or a couch fits through a door.

Navigation and travel

If you walk 3 blocks east and 4 blocks north, you are \(\sqrt{9 + 16} = 5\) blocks from your starting point in a straight line.

Construction

Builders use the “3-4-5 rule” to lay perfectly square corners. Measure 3 feet on one wall, 4 feet on the other; if the diagonal is exactly 5 feet, the corner is a right angle.

Common Mistakes

Using the wrong side as the hypotenuse

The hypotenuse is always opposite the right angle and is always the longest side. Confusing legs and hypotenuse breaks the formula.

Forgetting to square root at the end

\(c^2 = 100\) → \(c = 10\), not \(c = 100\).

Adding when you should subtract

To find a leg, subtract: \(a^2 = c^2 – b^2\).

Plugging into the distance formula wrong

\((x_2 – x_1)\), not \((x_1 – x_2)\) — though after squaring it doesn’t matter. Just stay organized.

Misidentifying right triangles

Don’t assume a triangle is a right triangle unless the problem says so or a small square symbol marks one corner.

Working without units

A ladder that is “13” long isn’t 13 — it’s 13 feet, 13 meters, or 13 cubits. Track units carefully.

How to Practice

1. Memorize the first 4 Pythagorean triples. Recognition is half the speed.
2. Drill finding hypotenuses until you can do 3-4-5 in your head.
3. Drill finding legs — these trip up more students than hypotenuses.
4. Mix in word problems — ladders, distance, diagonals.
5. Practice with the distance formula — it’s the same skill.

Free Resources

Effortless Math has a complete Pythagorean library:

• Geometry Worksheets — triangle and Pythagorean problems with answers.
• Math Topics Library — every geometry topic explained.
• Geometry eBooks — full geometry workbooks.

Frequently Asked Questions

Does the Pythagorean theorem work for any triangle?
No — only right triangles. For non-right triangles, use the Law of Cosines.

Who was Pythagoras?
A Greek mathematician from around 500 BC. The theorem was actually known before him (Babylonians, Egyptians, Indians) — he is credited with the first known proof.

How many proofs of the Pythagorean theorem are there?
Over 350. President Garfield (yes, the U.S. president) published one in 1876.

Is there a 3D version?
Yes — for a rectangular box with sides \(a\), \(b\), \(c\), the space diagonal is \(\sqrt{a^2 + b^2 + c^2}\).

Why do I keep seeing 3-4-5 triangles on tests?
Because they have clean integer sides. Test writers love them because the answers stay neat.

Can the legs be longer than the hypotenuse?
No. The hypotenuse is always the longest side of a right triangle.

You’re Set for Every Right Triangle Problem

The Pythagorean theorem is short, memorable, and shows up on every standardized test you’ll ever take. Memorize the formula. Memorize the triples. Practice finding both legs and hypotenuses. By next week, you will solve any Pythagorean problem in under a minute.

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