Hypotenuse Calculator
Calculate the hypotenuse of a right triangle using multiple methods
Calculate Hypotenuse
Vertical leg length
Horizontal leg length
Ladder Safety Example
Problem
A ladder needs to reach a 10 ft high roof. For safety, the ladder should be placed at a 75.5° angle with the ground. How long should the ladder be?
Solution
Given: Height (side a) = 10 ft, Angle α = 75.5°
Formula: c = a / sin(α)
Calculation: c = 10 / sin(75.5°) = 10 / 0.9703 = 10.31 ft
Result: The ladder should be approximately 10.31 feet long
Safety tip: Follow the 4:1 rule - for every 4 feet of height, place the ladder base 1 foot from the wall
Hypotenuse Formulas
Two Sides
Pythagorean theorem
Angle + Side
Trigonometric ratios
Area + Side
Using area formula
Quick Facts
Understanding the Hypotenuse
What is a Hypotenuse?
The hypotenuse is the longest side of a right triangle. It's the side opposite to the right angle (90°). The term comes from the ancient Greek word "hypoteinousa," meaning "stretching under a right angle."
Key Properties
- •Always the longest side in a right triangle
- •Opposite to the 90° angle
- •Length given by the Pythagorean theorem
- •Essential for trigonometric calculations
Calculation Methods
Pythagorean Theorem
Most common method when both legs are known
c² = a² + b²
Take the square root to find c
Trigonometric Ratios
When you know an angle and one side
sin(θ) = opposite/hypotenuse
cos(θ) = adjacent/hypotenuse
Real-World Applications
Construction
Ladder placement, roof angles, structural supports
Navigation
Distance calculations, GPS systems, surveying
Physics
Vector calculations, force resolution, projectile motion