How to Use the Pythagorean Theorem (With Real Examples)

The Pythagorean theorem is one of the most useful formulas in mathematics. It connects the three sides of any right triangle through a single elegant equation. Builders use it to square foundations. Pilots use it to calculate distances. Screen manufacturers use it to specify display sizes. Anyone who works with measurements in two or three dimensions will eventually reach for this theorem.

This guide covers the formula, visual proofs, common Pythagorean triples, and real-world applications with worked examples. For instant calculations, try our Pythagorean Theorem Calculator.

The Formula

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

$a^2 + b^2 = c^2$

The hypotenuse (c) is always the side opposite the 90-degree angle and always the longest side.

To find the hypotenuse:

$c = \sqrt{a^2 + b^2}$

To find a missing leg:

$a = \sqrt{c^2 - b^2}$

Example: A right triangle has legs of 6 and 8. Find the hypotenuse.

c = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10

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

a = sqrt(13^2 - 5^2) = sqrt(169 - 25) = sqrt(144) = 12

Visual Proof: Why It Works

The most intuitive proof uses squares built on each side of the triangle.

Proof by rearrangement:

Draw a right triangle with sides a, b, and c.
Build a square on each side. The areas of these squares are a^2, b^2, and c^2.
The square on the hypotenuse (c^2) has the same area as the other two squares combined (a^2 + b^2).

Picture a 3-4-5 triangle. The square on side 3 has 9 unit squares. The square on side 4 has 16 unit squares. The square on the hypotenuse 5 has 25 unit squares. And 9 + 16 = 25.

Proof by area (Garfield’s proof):

President James Garfield published this proof in 1876. Arrange two copies of the triangle and a third shape into a trapezoid. The trapezoid’s area calculated two different ways yields a^2 + b^2 = c^2.

Algebraic proof:

Place four identical right triangles inside a large square with side (a + b). The four triangles leave a tilted square of side c in the center.

Large square area: (a + b)^2 = a^2 + 2ab + b^2
Four triangles area: 4 x (1/2 x a x b) = 2ab
Inner square area: c^2
Since large square = four triangles + inner square: a^2 + 2ab + b^2 = 2ab + c^2
Cancel 2ab from both sides: a^2 + b^2 = c^2

There are over 370 known proofs of this theorem. The three above are the most commonly taught.

Pythagorean Triples

A Pythagorean triple is a set of three whole numbers that satisfy a^2 + b^2 = c^2. These are worth memorizing because they let you solve problems without a calculator.

Triple	Verification
3, 4, 5	9 + 16 = 25
5, 12, 13	25 + 144 = 169
8, 15, 17	64 + 225 = 289
7, 24, 25	49 + 576 = 625
9, 40, 41	81 + 1,600 = 1,681
20, 21, 29	400 + 441 = 841

Scaling property: Multiply every number in a triple by the same factor and you get another valid triple.

3-4-5 scaled by 2 = 6-8-10 (36 + 64 = 100)
3-4-5 scaled by 3 = 9-12-15 (81 + 144 = 225)
5-12-13 scaled by 2 = 10-24-26 (100 + 576 = 676)

Generating triples: For any two positive integers m and n where m > n:

$a = m^2 - n^2, \quad b = 2mn, \quad c = m^2 + n^2$

Using m = 2, n = 1: a = 3, b = 4, c = 5. Using m = 3, n = 2: a = 5, b = 12, c = 13.

Real-World Applications

Construction: Squaring Corners

The 3-4-5 method is the standard field technique for verifying that a corner is a true 90-degree angle. Every framer, mason, and concrete contractor uses it.

How to square a corner:

From the corner, measure 3 feet along one wall and mark it.
Measure 4 feet along the other wall and mark it.
The diagonal between the two marks should be exactly 5 feet.
If it isn’t 5 feet, the corner isn’t square. Adjust until it is.

For larger projects, use multiples: 6-8-10, 9-12-15, or 12-16-20. Bigger triangles give more accurate results because small measurement errors have less impact.

Example: You are laying out a 24 ft x 30 ft deck. To check the corner:

Measure 9 ft along the 24-ft side.
Measure 12 ft along the 30-ft side.
Diagonal should be exactly 15 ft (9-12-15 triple).

When you know the east-west distance and north-south distance between two points, the theorem gives you the direct “as the crow flies” distance.

Example: A hiker walks 4 miles east and then 3 miles north. How far is the hiker from the starting point in a straight line?

Distance = sqrt(4^2 + 3^2) = sqrt(16 + 9) = sqrt(25) = 5 miles

This also works for flight planning, boat navigation, and calculating GPS distances on a flat plane. For longer distances, the curvature of the earth requires adjustments, but for distances under 100 miles the flat approximation is accurate enough.

Screen Sizes: Diagonal Measurement

TV and monitor screens are measured diagonally. Manufacturers give you the diagonal size, but you often need the width and height for mounting brackets and furniture placement.

Example: A 55-inch TV has a 16:9 aspect ratio. What are the width and height?

The ratio means width = 16k and height = 9k for some value k.

Diagonal^2 = width^2 + height^2
55^2 = (16k)^2 + (9k)^2
3,025 = 256k^2 + 81k^2
3,025 = 337k^2
k^2 = 8.9762
k = 2.996
Width = 16 x 2.996 = 47.94 inches
Height = 9 x 2.996 = 26.96 inches

Ladders: Safe Placement

OSHA recommends a 4:1 ratio for ladder placement: for every 4 feet of height, the base should be 1 foot from the wall. The Pythagorean theorem tells you how long the ladder needs to be.

Example: You need to reach a gutter at 16 feet. The base of the ladder sits 4 feet from the wall.

Ladder length = sqrt(16^2 + 4^2) = sqrt(256 + 16) = sqrt(272) = 16.49 feet

You need at least a 17-foot ladder.

Roofing: Rafter Length

Roof pitch tells you the rise per 12 inches of run. A 6/12 pitch means the roof rises 6 inches for every 12 inches of horizontal run. The rafter follows the hypotenuse.

Example: A roof has a 6/12 pitch. The horizontal run from the exterior wall to the ridge is 14 feet. What is the rafter length?

Rise = 14 x (6/12) = 7 feet
Rafter = sqrt(14^2 + 7^2) = sqrt(196 + 49) = sqrt(245) = 15.65 feet

Add overhang length (typically 1-2 feet) to get the total rafter cut.

3D Distance

The theorem extends to three dimensions. To find the diagonal distance through a box:

$d = \sqrt{l^2 + w^2 + h^2}$

Example: Will a 9-foot fishing rod fit diagonally inside an SUV cargo area measuring 6 ft long, 4 ft wide, and 3 ft tall?

Diagonal = sqrt(36 + 16 + 9) = sqrt(61) = 7.81 feet

No. The rod won’t fit. You need to fold down seats or transport it on a roof rack.

Common Mistakes

1. Using the formula on non-right triangles. The Pythagorean theorem only works when one angle is exactly 90 degrees. For other triangles, use the law of cosines: c^2 = a^2 + b^2 - 2ab cos(C).

2. Assigning the hypotenuse to the wrong side. The hypotenuse is always opposite the right angle. If you label a leg as c, the formula will give you a wrong (and often impossible) answer.

3. Forgetting to square root. The formula gives you c^2, not c. A common error is to compute 3^2 + 4^2 = 25 and stop there. The hypotenuse is sqrt(25) = 5, not 25.

4. Mixing units. If one leg is in feet and the other in inches, convert them to the same unit before applying the formula.

Skip the Math

Our Pythagorean Theorem Calculator lets you enter any two sides of a right triangle and instantly find the third. It also identifies whether your inputs form a Pythagorean triple.

Frequently Asked Questions

Does the Pythagorean theorem work in three dimensions?

Yes. The 3D distance formula is a direct extension: d = sqrt(x^2 + y^2 + z^2). This gives you the straight-line distance between two points in space. It works for room diagonals, cable runs through buildings, and any situation where you need to find the shortest path through a rectangular volume.

What is the converse of the Pythagorean theorem?

If a^2 + b^2 = c^2 for three given side lengths, the triangle must be a right triangle. This is the converse, and it’s equally useful. Construction workers apply the converse when they measure three sides and check whether they satisfy the equation to verify a 90-degree angle.

Can the Pythagorean theorem use decimals or fractions?

The formula works with any positive numbers, not just whole numbers. A triangle with legs of 2.5 and 6 has a hypotenuse of sqrt(6.25 + 36) = sqrt(42.25) = 6.5. Pythagorean triples specifically refer to whole-number solutions, but the theorem itself has no such restriction.

Who was Pythagoras, and did he actually discover the theorem?

Pythagoras was a Greek philosopher and mathematician who lived around 570-495 BCE. However, Babylonian clay tablets from around 1800 BCE show that this relationship was known over 1,000 years before Pythagoras. Chinese and Indian mathematicians also documented the theorem independently. Pythagoras (or his school) is credited with providing the first known formal proof.

How accurate is the 3-4-5 method for squaring corners?

It is very accurate when done carefully. The main sources of error are imprecise tape measure readings and sagging tapes over long distances. Using larger multiples (like 12-16-20) reduces the percentage impact of small measurement errors. For professional work, combine the 3-4-5 check with a diagonal measurement across the full rectangle: both diagonals should be equal if all corners are square.