a and b
The legs meet to form the right angle. Either leg may be labeled a or b.
Free Geometry Guide
The Pythagorean Theorem
Learn how to identify the hypotenuse, choose the correct equation, find missing side lengths, simplify radical answers, calculate coordinate-plane distance, and model real-world problems.
The Pythagorean theorem describes the relationship among the three sides of a right triangle. When two side lengths are known, the theorem can be used to find the third. The key is to identify the hypotenuse correctly before substituting any numbers.
What Is the Pythagorean Theorem?
In every right triangle, the sum of the squares of the two leg lengths equals the square of the hypotenuse length. The theorem is written using a and b for the legs and c for the hypotenuse.
a² + b² = c²
When Can You Use It?
Use the Pythagorean theorem only when the triangle contains a 90° angle. The right-angle marker is usually drawn as a small square.
The Legs and the Hypotenuse
c
The hypotenuse is opposite the right angle and is always the longest side.
How to Identify the Hypotenuse
1. Find the right angle
Locate the 90° marker where the two legs meet.
2. Look across the triangle
The side directly opposite the right angle is the hypotenuse.
3. Label it c
The hypotenuse must occupy the c position in a² + b² = c².
Do Not Identify It by Position
A rotated triangle may place the hypotenuse on the left, right, top, or bottom. Use the right angle—not the picture’s orientation.
How to Find the Hypotenuse
When both legs are known and c is missing, square the two legs, add the results, and take the positive square root.
Example: a = 6 and b = 8
Write the formula
a² + b² = c²
Substitute
6² + 8² = c²
Square
36 + 64 = c²
Add
100 = c²
Take the square root
c = √100 = 10
The hypotenuse is 10 units.
How to Find a Missing Leg
When the hypotenuse and one leg are known, keep the hypotenuse in the c position. Subtract the known leg’s square from the hypotenuse’s square.
Example: b = 12 and c = 13
Write the formula
a² + b² = c²
Substitute
a² + 12² = 13²
Square
a² + 144 = 169
Subtract
a² = 169 − 144 = 25
Take the square root
a = √25 = 5
The Subtraction Order Matters
Use hypotenuse squared minus known leg squared. Because c is the longest side, c² must be the larger value.
Exact Radicals and Rounded Decimals
| Answer type | What to do | Example for √72 |
|---|---|---|
| Exact radical | Factor out the greatest perfect-square factor. | √72 = √(36 · 2) = 6√2 |
| Nearest tenth | Use a calculator, then round to one decimal place. | √72 ≈ 8.5 |
| Nearest hundredth | Use a calculator, then round to two decimal places. | √72 ≈ 8.49 |
Side Lengths Are Positive
Although an equation such as x² = 25 has two algebraic solutions, a geometric side length uses the positive value: x = 5.
Using the Pythagorean Theorem for Coordinate Distance
The horizontal and vertical changes between two points form the legs of a right triangle. The straight-line distance between the points is the hypotenuse.
Example: (−2, 1) and (4, 9)
Horizontal change
Δx = |4 − (−2)| = 6
Vertical change
Δy = |9 − 1| = 8
Use the theorem
6² + 8² = d²
Solve
36 + 64 = 100, so d = 10
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Modeling Pythagorean Theorem Word Problems
Wall, Ground, and Ladder
The wall and ground are the legs. The ladder is the hypotenuse.
Pole, Ground, and Wire
The pole and ground distance are the legs. The wire is the hypotenuse.
Length, Width, and Diagonal
The length and width are the legs. The corner-to-corner diagonal is the hypotenuse.
Changes and Distance
Δx and Δy are the legs. The direct distance between the points is the hypotenuse.
Draw and Label Before Solving
Decide which measurement is the hypotenuse before placing any value into the equation.
Worked Pythagorean Theorem Examples
Legs 9 and 12
- 9² + 12² = c²
- 81 + 144 = 225
- c = √225 = 15
Leg 7, Hypotenuse 9
- a² + 7² = 9²
- a² = 81 − 49 = 32
- a = √32 = 4√2
Legs 5 and 8
- 5² + 8² = c²
- c² = 89
- c = √89 ≈ 9.4
13-Foot Ladder
- The ladder is c = 13.
- The ground distance is 5.
- Height = √(13² − 5²) = 12 feet.
8 by 15 Rectangle
- 8² + 15² = d²
- d² = 64 + 225 = 289
- d = 17 units
Changes 3 and 4
- Δx = 3 and Δy = 4.
- 3² + 4² = d²
- d = 5 units
How to Check a Missing-Side Answer
Place the completed side lengths back into a² + b² = c². The two sides of the equation must be equal, or approximately equal when a side was rounded.
Verify a 5–12–13 Triangle
Left side
5² + 12² = 25 + 144 = 169
Right side
13² = 169
Compare
169 = 169, so the side lengths satisfy the theorem.
Common Pythagorean Theorem Mistakes
Using the Theorem on a Non-Right Triangle
The relationship a² + b² = c² requires a 90° angle.
Calling the Longest-Looking Side c
Use the side opposite the right angle. Do not rely on how the drawing is rotated or scaled.
Putting a Leg in the c Position
The hypotenuse is the only side that belongs by itself on the right side of the formula.
Adding When a Leg Is Missing
To find a leg, subtract the known leg square from the hypotenuse square.
Subtracting in the Wrong Order
Calculate c² minus the known leg square so the missing square remains positive.
Forgetting the Square Root
An equation such as c² = 100 is not finished until you write c = 10.
Using a Negative Side Length
Geometric lengths use the positive square root.
Rounding Too Early
Keep the square-root expression or full calculator value until the final step.
A Reliable Pythagorean Theorem Strategy
1. Confirm a right triangle
Look for a 90° angle or a context that creates perpendicular sides.
2. Identify the hypotenuse
Find the side opposite the right angle and label it c.
3. Label the legs
Use a and b for the two sides that form the right angle.
4. Write the formula
Begin with a² + b² = c² before substituting values.
5. Substitute and square
Place each known length in the correct position and evaluate its square.
6. Add or subtract
Add to find c; subtract from c² to find a missing leg.
7. Take the square root
Use the positive root and simplify or round as requested.
8. Verify
Substitute all three completed side lengths into the theorem.
Practice the Pythagorean Theorem
Work through missing hypotenuses, missing legs, coordinate-plane distances, and real-world problems one typed step at a time.
Open the Pythagorean Theorem ToolRelated Geometry Resources
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