The Pythagorean Theorem with Real Examples

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse's length equals the sum of the squares of the two other sides' lengths. NASA's Glenn Research Center describes this directly: the area of the square drawn on the hypotenuse equals the area of the square drawn on one leg plus the area of the square drawn on the other leg. Written as a formula, with c as the hypotenuse and a and b as the two legs, this is a squared plus b squared equals c squared.

This only applies to right triangles, meaning triangles with one 90-degree angle. The hypotenuse is always the longest side, opposite the right angle. Below are four worked examples, each with the arithmetic double-checked.

Example 1: The classic 3-4-5 triangle

If a right triangle has legs of length 3 and 4, what is the length of the hypotenuse?

Step 1: Square both legs. 3 squared is 9, and 4 squared is 16.

Step 2: Add the squares together. 9 + 16 = 25.

Step 3: Take the square root of the sum to find the hypotenuse. The square root of 25 is 5.

So the hypotenuse is 5. This is why the 3-4-5 triangle is often used as a first example: all three sides come out as whole numbers, which makes it easy to verify by hand.

Example 2: Finding a missing leg

If a right triangle has a hypotenuse of 13 and one leg of 5, what is the length of the other leg?

This time you're solving for a leg instead of the hypotenuse, which means rearranging the formula. Instead of adding the two legs' squares, you subtract the known leg's square from the hypotenuse's square.

Step 1: Square the hypotenuse and the known leg. 13 squared is 169, and 5 squared is 25.

Step 2: Subtract the known leg's square from the hypotenuse's square. 169 - 25 = 144.

Step 3: Take the square root. The square root of 144 is 12.

The missing leg is 12. Notice that 5, 12, and 13 is another set of whole numbers that satisfies the theorem, similar to the 3-4-5 triangle.

Example 3: A ladder leaning against a wall

A 10-foot ladder is leaning against a wall, with its base 6 feet from the wall. How high up the wall does the ladder reach?

This is a real-world application where the ladder itself is the hypotenuse, the distance from the wall to the base is one leg, and the height up the wall is the other leg, the one you're solving for.

Step 1: Square the ladder's length and the base distance. 10 squared is 100, and 6 squared is 36.

Step 2: Subtract. 100 - 36 = 64.

Step 3: Take the square root. The square root of 64 is 8.

The ladder reaches 8 feet up the wall. This is also a 6-8-10 triangle, which is just the 3-4-5 triangle with every side doubled, a useful pattern to recognize since scaled versions of the same ratio also satisfy the theorem.

Example 4: A rectangular park's diagonal path

A rectangular park measures 120 meters by 90 meters. If a path cuts diagonally from one corner to the opposite corner, how long is that path?

Here, the two sides of the rectangle act as the two legs of a right triangle, and the diagonal path is the hypotenuse.

Step 1: Square both sides. 120 squared is 14,400, and 90 squared is 8,100.

Step 2: Add. 14,400 + 8,100 = 22,500.

Step 3: Take the square root. The square root of 22,500 is 150.

The diagonal path is 150 meters long. This example shows the theorem doesn't only apply to small triangles on paper; it works at any scale, which is part of why it shows up in real surveying and construction problems.

A common mistake to avoid

A frequent error is adding the hypotenuse's square to one leg's square instead of subtracting, when solving for a missing leg like in Example 2 and Example 3. Remember: the hypotenuse's square is always the largest value in the equation, since it's the sum of both legs' squares. If you're solving for a leg, you subtract the known leg's square from the hypotenuse's square, not the other way around.

Why this theorem matters beyond triangles

According to the Common Core State Standards Initiative's Grade 8 mathematics overview, applying the Pythagorean theorem to find distances between points on a coordinate plane is a standard part of Grade 8 geometry. This connects directly to Example 4 above: finding a straight-line distance between two points, like opposite corners of a rectangle, is really the same calculation as finding a hypotenuse, just applied to coordinates instead of a drawn triangle.

Key takeaways

• The Pythagorean theorem states a squared plus b squared equals c squared, where c is the hypotenuse of a right triangle.
• To find the hypotenuse, add the squares of both legs, then take the square root.
• To find a missing leg, subtract the known leg's square from the hypotenuse's square, then take the square root.
• Scaled versions of known triangles, like 3-4-5 and 6-8-10, satisfy the theorem at any multiple of the original ratio.
• The same method applies to real-world problems, like ladder heights and diagonal paths, not just triangles drawn on paper.