Pythagorean Theorem

The Pythagorean theorem is one of the most useful formulas in all of mathematics. It describes the relationship between the three sides of a right triangle — a triangle that has one 90-degree angle.

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

Parts of a Right Triangle

Every right triangle has three sides:

• Legs ($a$ and $b$): the two sides that form the right angle
• Hypotenuse ($c$): the longest side, directly across from the right angle

The hypotenuse is always the longest side. It is always opposite the 90-degree angle.

Using the Theorem to Find a Missing Side

Finding the Hypotenuse

When you know both legs, solve for $c$:

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

Example 1: A right triangle has legs of 6 and 8. Find the hypotenuse.

$c = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$

Answer: The hypotenuse is 10.

Finding a Leg

When you know the hypotenuse and one leg, rearrange the formula to solve for the missing leg:

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

Example 2: A right triangle has a hypotenuse of 13 and one leg of 5. Find the other leg.

$a = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$

Answer: The missing leg is 12.

Example 3: A 20-foot ladder leans against a wall. The base of the ladder is 8 feet from the wall. How high up the wall does the ladder reach?

The ladder is the hypotenuse ($c = 20$), the distance from the wall is one leg ($b = 8$), and the height up the wall is the unknown leg ($a$).

$a = \sqrt{20^2 - 8^2} = \sqrt{400 - 64} = \sqrt{336} \approx 18.33 \text{ ft}$

Answer: The ladder reaches approximately 18.33 feet up the wall.

Common Pythagorean Triples

A Pythagorean triple is a set of three whole numbers that satisfy $a^2 + b^2 = c^2$. Memorizing these saves time — if you recognize the pattern, you don’t need to calculate.

Multiples of triples also work. The 3-4-5 triple scales to 6-8-10, 9-12-15, 12-16-20, and so on. Any time you multiply all three numbers by the same factor, the result is still a valid triple.

The 3-4-5 Rule for Squaring Corners

The 3-4-5 triple has a special place in the trades. Carpenters, concrete workers, and framers use it every day to check whether a corner forms a perfect 90-degree angle.

The idea is simple: if you measure 3 feet along one side, 4 feet along the other side, and the diagonal between those two points is exactly 5 feet, the corner is square. If the diagonal is off, the corner isn’t 90 degrees.

Any multiple works — 6-8-10 or 9-12-15 gives you a larger triangle and a more accurate check over longer distances.

Real-World Application: Carpentry — Squaring a Wall Corner

A carpenter is framing a room and needs to verify that the corner where two walls meet is a perfect 90-degree angle. Here is the 3-4-5 method:

Step 1 — Measure 3 feet along the bottom plate of one wall from the corner and make a mark.

Step 2 — Measure 4 feet along the bottom plate of the other wall from the corner and make a mark.

Step 3 — Measure the diagonal between the two marks.

Step 4 — Check the result:

$3^2 + 4^2 = 9 + 16 = 25 = 5^2$

If the diagonal measures exactly 5 feet, the corner is square. If it’s longer than 5 feet, the angle is greater than 90 degrees (the walls are too far apart). If it’s shorter than 5 feet, the angle is less than 90 degrees (the walls are too close together).

For greater accuracy on longer walls, the carpenter can use the 6-8-10 version:

$6^2 + 8^2 = 36 + 64 = 100 = 10^2$

Why this matters: An out-of-square wall causes problems throughout the entire build — flooring won’t line up, cabinets won’t fit, and drywall will have visible gaps.

When the Answer Isn’t a Whole Number

Not every right triangle produces clean whole numbers. When the result is an irrational number, round to a practical level of precision.

Example 4: A right triangle has legs of 5 and 7. Find the hypotenuse.

$c = \sqrt{5^2 + 7^2} = \sqrt{25 + 49} = \sqrt{74} \approx 8.60$

Answer: The hypotenuse is approximately 8.60.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• The Pythagorean theorem applies only to right triangles: $a^2 + b^2 = c^2$
• The hypotenuse ($c$) is always the longest side, opposite the right angle
• To find the hypotenuse: $c = \sqrt{a^2 + b^2}$
• To find a missing leg: $a = \sqrt{c^2 - b^2}$
• Pythagorean triples (3-4-5, 5-12-13, 8-15-17) and their multiples give whole-number answers
• The 3-4-5 rule is used in the trades daily to check if corners are square

Return to Geometry for more topics in this section.