Pythagorean Theorem Calculator
Calculate missing sides of right triangles using the Pythagorean theorem: a² + b² = c²
Calculate Missing Side
Hypotenuse (c)
Given legs a and b
c = √(a² + b²)
Leg (a)
Given leg b and hypotenuse c
a = √(c² - b²)
Leg (b)
Given leg a and hypotenuse c
b = √(c² - a²)
Please enter both legs (a and b) to calculate the hypotenuse.
Calculation Results
Example: Ladder Problem
Real-world Application
Problem: A ladder is leaning against a wall. The wall is 4 meters high, and the base of the ladder is 3 meters away from the wall. How long is the ladder?
Given: Leg a (height) = 4 m, Leg b (distance) = 3 m
Find: Hypotenuse c (ladder length)
Solution
Step 1: Apply Pythagorean theorem: c² = a² + b²
Step 2: Substitute values: c² = 4² + 3² = 16 + 9 = 25
Step 3: Take square root: c = √25 = 5 meters
Answer: The ladder is 5 meters long.
Pythagorean Theorem
Basic Formula
a² + b² = c²
Where c is the hypotenuse (longest side)
a, b: Legs of the right triangle
c: Hypotenuse (opposite the right angle)
Note: Only applies to right triangles (90° angle)
Common Pythagorean Triples
Quick Tips
The hypotenuse is always the longest side
Only works for right triangles (90° angle)
All side lengths must be positive
The hypotenuse must be longer than either leg
Understanding the Pythagorean Theorem
What is the Pythagorean Theorem?
The Pythagorean theorem is a fundamental mathematical principle that describes the relationship between the three sides of a right triangle. It states that the sum of the squares of the legs equals the square of the hypotenuse.
Historical Background
Named after the ancient Greek mathematician Pythagoras (c. 570-495 BC), though the relationship was known to earlier civilizations including the Babylonians and Indians. Pythagoras is credited with providing the first formal proof.
Real-world Applications
- •Construction and architecture for measuring distances
- •Navigation and GPS systems
- •Engineering and surveying
- •Physics and trigonometry
Three Solution Methods
Finding the Hypotenuse
When you know both legs (a and b):
c = √(a² + b²)
Finding a Leg
When you know one leg and the hypotenuse:
a = √(c² - b²)
b = √(c² - a²)
Important Notes
- • The triangle must have a 90° angle
- • The hypotenuse is opposite the right angle
- • All measurements must use the same units
- • Results are always positive values