The Ultimate Guide to the Pythagorean Theorem
The Pythagorean theorem is a cornerstone of geometry, describing the relationship between the three sides of a right triangle. Named after the ancient Greek mathematician Pythagoras, this theorem is fundamental not only in mathematics but also in numerous real-world applications, especially in construction and engineering. This guide explains the theorem, its formulas, and how our calculator can solve for any missing side or angle, with steps included.
What is the Pythagorean Theorem Formula?
For any right triangle, where 'a' and 'b' are the lengths of the two legs (the sides that form the right angle) and 'c' is the length of the hypotenuse (the longest side, opposite the right angle), the formula is:
a² + b² = c²
This simple equation allows you to find the length of any side if you know the lengths of the other two.
How to Find Side a, b, or c
The formula can be rearranged to solve for any of the sides:
- To find the hypotenuse (c):
c = √(a² + b²) - To find a leg (a):
a = √(c² - b²) - To find a leg (b):
b = √(c² - a²)
Our calculator performs these calculations for you and shows the step-by-step work.
How to Use the Calculator
Our tool is designed for ease of use and provides a complete solution.
- Enter Two Sides: Input the lengths of the two sides you know into their respective fields (a, b, or c). Leave the unknown side's field blank.
- Select Units: Choose your unit of measurement. If you're working with feet and inches, convert your measurement to a single unit first (e.g., 5' 6" becomes 66 inches).
- Click "Calculate": The tool will instantly compute all missing values.
- Review the Results: You will get the length of the missing side, the two acute angles (α and β) in degrees, the triangle's area, and its perimeter.
Finding the Angles of a Right Triangle
Our calculator also finds the two missing angles using trigonometry.
- Angle α (opposite side a) is found using
α = arcsin(a/c). - Angle β (opposite side b) is found using
β = arcsin(b/c).
The result is given in degrees, providing a complete picture of your triangle's geometry.
Practical Application in Construction
The Pythagorean theorem is essential in construction for ensuring that corners are perfectly square (90 degrees). Builders use a method called the "3-4-5 rule," which is based on the Pythagorean triple 3² + 4² = 5² (9 + 16 = 25).
To square a corner, a builder will measure 3 feet along one wall and 4 feet along the adjacent wall. The distance between those two points must be exactly 5 feet for the corner to be a perfect right angle. This simple application of the theorem is used daily on job sites around the world.
Frequently Asked Questions (FAQ)
How does the calculator find a missing side, like side b?
To find a missing leg (like side b), the calculator rearranges the Pythagorean theorem to b² = c² - a². It then takes the square root to find the length of b. The full calculation is shown in the 'steps' section.
Can this calculator find the angles?
Yes. After calculating all three sides, the calculator uses trigonometric functions (like arcsin and arccos) to find the two acute angles of the right triangle, displaying them in degrees.
How do I use this for feet and inches?
To use the calculator with feet and inches, first convert your measurement to a single unit. For example, 5 feet 6 inches becomes 66 inches (since 5 * 12 + 6 = 66). Enter this value and select 'inches' as your unit.
For more triangle-related calculations, check out our Area Calculator. To understand the steepness of the hypotenuse, see our Slope Calculator.