Pythagoras' Theorem Calculator
Calculate the hypotenuse of a right-angled triangle.
Formula first
Overview
Pythagoras' Theorem establishes a fundamental relationship between the three sides of a right-angled triangle in Euclidean geometry. It states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Symbols
Variables
a = Side a, b = Side b, c = Hypotenuse
Apply it well
When To Use
When to use: Apply this theorem when you need to calculate an unknown side of a triangle that contains a 90-degree angle. It is valid in flat, two-dimensional space and assumes the triangle is perfectly right-angled.
Why it matters: This principle is the cornerstone of trigonometry and coordinate geometry, enabling precise distance calculations in GPS technology and architecture. It allows engineers to ensure structures are stable and square, and helps physicists calculate the magnitude of vectors.
Avoid these traps
Common Mistakes
- Forgetting to square root the final answer.
- Adding squares when finding a shorter side.
One free problem
Practice Problem
Practice Problem 1
A park designer is creating a rectangular garden path. If one side of the garden is 6 meters long and the adjacent side is 8 meters long, what is the length of the diagonal path connecting the opposite corners?
Solve for:
Hint: Square both side lengths, add them together, then find the square root of the sum.
Practice Problem 2
An architect is designing a roof truss in the shape of a right triangle. The sloping side is 13 feet long, and the horizontal base is 12 feet long. What is the vertical height of the truss?
Solve for:
Hint: Rearrange the formula to a² = c² - b² to isolate the missing leg.
Practice Problem 3
A hiker travels 9 kilometers due north and then turns to travel a certain distance due east. If the direct distance from the starting point to the current location is 15 kilometers, how many kilometers did the hiker travel east?
Solve for:
Hint: The direct distance (15 km) represents the hypotenuse c.
The full worked solution stays in the interactive walkthrough.
References
Sources
- Wikipedia: Pythagorean theorem
- Britannica: Pythagorean theorem
- Halliday, Resnick, Walker, Fundamentals of Physics
- Wikipedia: Pythagoras' theorem
- Euclid's Elements, Book I, Proposition 47
- Standard curriculum — GCSE Maths (Geometry)