Pythagorean Theorem Guide: Master Right Triangles with Easy Examples

The Pythagorean theorem is one of the most famous and useful formulas in all of mathematics. For over 2,500 years, this elegant relationship has helped people solve problems involving right triangles—from ancient architects building pyramids to modern engineers designing skyscrapers. This complete guide shows you how it works, why it matters, and how to use it with confidence using MathAI GPT.

1) What is the Pythagorean Theorem?

The Pythagorean theorem describes the relationship between the three sides of a right triangle (a triangle with one 90-degree angle). It states that the square of the hypotenuse equals the sum of the squares of the other two sides.

The Pythagorean Theorem Formula

a² + b² = c²

Where:

a and b = the two shorter sides (legs) of the right triangle
c = the longest side (hypotenuse), opposite the right angle

Visual tip: Draw a right triangle. The hypotenuse is always the side opposite the 90° angle—it's the longest side and it never touches the right angle.

2) How to Use the Pythagorean Theorem

Finding the Hypotenuse (c)

When you know both legs, you can find the hypotenuse.

Example 1: Find the hypotenuse

Problem: A right triangle has legs of 3 and 4. Find the hypotenuse.

Step 1: Write the formula

a² + b² = c²

Step 2: Plug in the values

3² + 4² = c²

9 + 16 = c²

25 = c²

Step 3: Take the square root

c = √25 = 5

Famous triple! (3, 4, 5) is the most common Pythagorean triple—memorize it!

Finding a Leg (a or b)

When you know the hypotenuse and one leg, you can find the other leg by rearranging the formula.

Example 2: Find a missing leg

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

Step 1: Write the formula

a² + b² = c²

Step 2: Plug in known values (let a = 5, c = 13)

5² + b² = 13²

25 + b² = 169

Step 3: Solve for b²

b² = 169 - 25

b² = 144

Step 4: Take the square root

b = √144 = 12

Another famous triple! (5, 12, 13) appears frequently in math problems

3) Common Pythagorean Triples

Pythagorean triples are sets of three whole numbers that satisfy a² + b² = c². Memorizing these saves time!

Most Common Pythagorean Triples

Basic Triples

(3, 4, 5)
(5, 12, 13)
(8, 15, 17)
(7, 24, 25)

Multiples (×2)

(6, 8, 10)
(10, 24, 26)
(9, 12, 15)
(12, 16, 20)

Pro tip: Any multiple of a Pythagorean triple is also a triple. If (3, 4, 5) works, so does (6, 8, 10), (9, 12, 15), etc.

4) Real-World Applications

Construction and Carpentry

Example: Checking if a Corner is Square

Problem: A carpenter needs to verify a 90° corner. They measure 3 feet on one side and 4 feet on the other. What should the diagonal be?

Solution: 3² + 4² = c² → 9 + 16 = 25 → c = 5 feet

Carpenter's trick: The 3-4-5 rule! If the diagonal is exactly 5 feet, the corner is perfectly square (90°). Scale it up: 6-8-10, 9-12-15, etc.

Ladder Safety

Example: Safe Ladder Placement

Problem: You need to reach a window 12 feet high. For safety, the ladder base should be 5 feet from the wall. How long must the ladder be?

Solution:

5² + 12² = c²

25 + 144 = 169

c = √169 = 13 feet

Another 5-12-13 triple! Safety inspectors love this ratio.

Navigation and Distance

Example: Finding Shortest Distance

Problem: You walk 6 blocks east and 8 blocks north. What's the straight-line distance back to your starting point?

Solution:

6² + 8² = c²

36 + 64 = 100

c = √100 = 10 blocks

This is double the 3-4-5 triple (6-8-10)!

TV and Screen Sizes

Example: Actual Screen Size

Problem: A “50-inch TV” is measured diagonally. If the screen is 44 inches wide, how tall is it?

Solution:

44² + b² = 50²

1936 + b² = 2500

b² = 564

b = √564 ≈ 23.7 inches tall

TVs are measured by diagonal, but you need the Pythagorean theorem to find actual dimensions!

5) Advanced: 3D Distance Formula

The Pythagorean theorem extends to three dimensions! To find distance in 3D space:

3D Distance Formula

d² = x² + y² + z²

Example: Find the distance from (0,0,0) to (3,4,12)

d² = 3² + 4² + 12²

d² = 9 + 16 + 144 = 169

d = √169 = 13 units

6) Common Mistakes to Avoid

Using the wrong side as c: c is always the hypotenuse (longest side, opposite the right angle)
Forgetting to square root: After solving for c², you must take √c² to find c
Only works for right triangles: The theorem ONLY applies to triangles with a 90° angle
Negative answers: Side lengths are always positive; if you get a negative, check your math
Calculator in wrong mode: Make sure your calculator is in degree mode for trigonometry problems

7) Verifying a Right Triangle

You can use the Pythagorean theorem backwards to check if a triangle is a right triangle!

Test: Is This a Right Triangle?

Given sides: 7, 24, 25

Check: Does a² + b² = c²?

7² + 24² = 25²?

49 + 576 = 625?

625 = 625 ✓

Yes! This is a right triangle.

Rule: If the equation is true, it's a right triangle. If not, it's not!

8) Practice Problems

Find the hypotenuse of a right triangle with legs 5 and 12
A right triangle has a hypotenuse of 10 and one leg of 6. Find the other leg.
Is a triangle with sides 9, 40, 41 a right triangle?
A 25-foot ladder leans against a wall, with its base 7 feet from the wall. How high up does it reach?
You walk 9 km east, then 12 km north. What's the straight-line distance back?
A rectangular TV is 32 inches wide and 18 inches tall. What's the diagonal (advertised size)?
Find the missing side: legs are 8 and 15, find the hypotenuse
A baseball diamond is a square with 90-foot sides. What's the distance from home plate to second base?

Show answers

1. 5² + 12² = c² → 25 + 144 = 169 → c = 13

2. 6² + b² = 10² → 36 + b² = 100 → b² = 64 → b = 8

3. 9² + 40² = 81 + 1600 = 1681; 41² = 1681 → Yes, right triangle!

4. 7² + h² = 25² → 49 + h² = 625 → h² = 576 → h = 24 feet

5. 9² + 12² = c² → 81 + 144 = 225 → c = 15 km

6. 32² + 18² = d² → 1024 + 324 = 1348 → d ≈ 36.7 inches

7. 8² + 15² = c² → 64 + 225 = 289 → c = 17 (8-15-17 triple!)

8. 90² + 90² = c² → 8100 + 8100 = 16200 → c ≈ 127.3 feet (90√2)

Quick Reference

Formula: a² + b² = c²

Find hypotenuse: c = √(a² + b²)

Find leg: a = √(c² - b²)

Check right triangle: Does a² + b² = c²?

Remember: c is always the longest side (hypotenuse)

Memorize: (3,4,5), (5,12,13), (8,15,17), and their multiples

For complex problems or to verify your work, use a Pythagorean theorem calculator. For more triangle problems, try the right triangle calculator.

Next Steps

The Pythagorean theorem is a cornerstone of geometry and trigonometry. Once you master it, you'll see right triangles everywhere—and you'll know exactly how to solve them!

Ready to practice more? Head to MathAI GPT and paste in any Pythagorean theorem or right triangle problem. Get instant step-by-step solutions with visual diagrams, plus practice problems to build your skills. From basic homework to advanced applications, we make geometry easy to understand!