Pythagorean Theorem

beginnergeometrytriangles

Relates the three sides of a right triangle

The Formula

a² + b² = c²

Variables & Symbols

a, b

Lengths of the two legs

c

Length of the hypotenuse

When to Use This Formula

  • Finding the third side of a right triangle
  • Checking if a triangle is a right triangle
  • Distance calculations in 3D space
  • Navigation and surveying

Worked Examples

Example 1

Problem:

Find hypotenuse if a=3 and b=4

Solution: c = 5

Step-by-Step:

1

Apply theorem: 3² + 4² = c²

2

Calculate: 9 + 16 = c²

3

c² = 25

4

c = √25 = 5

Common Mistakes to Avoid

  • Using it for non-right triangles
  • Forgetting to take the square root
  • Confusing legs with hypotenuse

Historical Context

Named after ancient Greek mathematician Pythagoras (570-495 BC), though evidence suggests Babylonians knew it 1000 years earlier.

Real-World Applications

1

Construction - ensuring corners are square

2

Navigation - finding shortest distances

3

Computer graphics - 3D rendering

4

Astronomy - calculating distances

Practice Problems

Problem 1: If a=5 and b=12, find c

Show Answer

Answer: c = 13

Problem 2: If a=8 and c=10, find b

Show Answer

Answer: b = 6

Related Formulas

distance formula

View Formula →

triangle area

View Formula →

law of cosines

View Formula →

Tags:

geometrytrianglesright trianglepythagorean

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