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 \(c\):

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

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?

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

\(10^2 = 100\).

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 (resource currently unavailable), triangle and Pythagorean problems with answers.
• Math Topics Library, every geometry topic explained.
• Geometry eBooks, full geometry workbooks.

Frequently Asked Questions

How do you do the Pythagorean theorem? Label the sides first: the hypotenuse c is the side opposite the right angle, and a and b are the legs. Then use a² + b² = c². To find the hypotenuse, add the squares of the legs and take the square root. To find a leg, subtract instead: a² = c² − b². For a hypotenuse of 10 and a leg of 6, a² = 100 − 36 = 64, so the missing leg is 8.

What is the Pythagorean theorem formula? The formula is a² + b² = c², where a and b are the legs of a right triangle and c is the hypotenuse, the longest side. It only works for right triangles. Rearranged, it becomes c = √(a² + b²) for the hypotenuse or a = √(c² − b²) for a missing leg. In picture form, the square on the hypotenuse has the same area as the squares on the two legs combined.

How can you learn the Pythagorean theorem easily? Start with the picture: the squares on the two legs together match the area of the square on the hypotenuse. Next, memorize a few Pythagorean triples, such as 3-4-5, 5-12-13, 8-15-17 and 7-24-25, so you can spot answers quickly on tests. Practice both directions, finding the hypotenuse and finding a leg, and always check that your hypotenuse comes out as the longest side.

What are some Pythagorean theorem examples? With legs of 5 and 12, c² = 25 + 144 = 169, so the hypotenuse is 13. With a hypotenuse of 17 and one leg of 15, the other leg is √(289 − 225) = √64 = 8. In everyday life, a 13-foot ladder with its base 5 feet from a wall reaches 12 feet up the wall, because 5² + 12² = 13².

What is Pythagorean Theorem Day? Pythagorean Theorem Day is a date whose month, day and two-digit year form a Pythagorean triple. October 24, 2026 (10/24/26) is one, because 10² + 24² = 100 + 576 = 676 = 26². Earlier examples include 3/4/05, 6/8/10, 5/12/13, 8/15/17 and 7/24/25. Written in month/day/year form, no other date fits the pattern for the rest of this century.

How does a Pythagorean theorem calculator work? A Pythagorean theorem calculator takes the two sides you know and solves a² + b² = c² for the missing one. Enter both legs to get the hypotenuse, or enter the hypotenuse and one leg to get the other leg. It is handy for checking homework, but work a few by hand as well: for legs of 8 and 15, 64 + 225 = 289 and √289 = 17.

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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