📏 Pythagorean Theorem Calculator
Find missing sides of right triangles using a² + b² = c²
Find Hypotenuse (c)
Find Side a
Find Side b
Check Right Triangle
How to Use This Calculator
- Choose which side you need to find from the four calculator options
- To find the hypotenuse (c): Enter the lengths of both legs (a and b)
- To find a leg (a or b): Enter the hypotenuse (c) and the other known leg
- The calculator computes results instantly as you type with 4 decimal place precision
- Review the formula shown below each result to understand the calculation
- Use "Check Right Triangle" to verify if three sides satisfy a² + b² = c²
- All measurements should use the same unit (inches, centimeters, meters, etc.)
Understanding the Pythagorean Theorem
What is the Pythagorean Theorem?
The Pythagorean theorem is a fundamental principle in geometry discovered by the ancient Greek mathematician Pythagoras around 500 BCE. It states: a² + b² = c², 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). This relationship holds true for ALL right triangles—triangles with exactly one 90-degree angle. For example, in a right triangle with legs of 3 and 4 units, the theorem predicts the hypotenuse: c = √(3² + 4²) = √(9 + 16) = √25 = 5. This 3-4-5 triangle is the most famous Pythagorean triple.
How to Apply the Formula
The theorem can be rearranged to solve for any of the three sides. Finding the hypotenuse: c = √(a² + b²). Square each leg, add them, then take the square root. Example: If a=6 and b=8, then c = √(36+64) = √100 = 10. Finding a leg: a = √(c² - b²) or b = √(c² - a²). Square the hypotenuse, subtract the square of the known leg, then take the square root. Example: If c=13 and b=5, then a = √(169-25) = √144 = 12. Important: The hypotenuse (c) must always be the largest value. If you get a negative number under the square root, check your inputs—the hypotenuse should be larger than either leg.
Pythagorean Triples and Special Triangles
Pythagorean triples are sets of three positive integers (whole numbers) that satisfy a² + b² = c². Knowing these makes mental math easier. Common primitive triples (where the numbers share no common factor): (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), and (20, 21, 29). Any multiple of these is also a triple: (6, 8, 10), (9, 12, 15), (15, 20, 25), etc. Special case: The 45-45-90 triangle (isosceles right triangle) has legs of equal length; if each leg is 1, the hypotenuse is √2 ≈ 1.414. The 30-60-90 triangle has sides in the ratio 1 : √3 : 2. These patterns are invaluable in engineering, architecture, and navigation.
Real-World Applications
The Pythagorean theorem has countless practical uses. Construction & Carpentry: To ensure a corner is perfectly square (90°), builders use the 3-4-5 rule—measure 3 feet along one edge, 4 feet along the perpendicular edge, and the diagonal should be exactly 5 feet. Navigation & Distance: Calculate straight-line distance between two points on a grid. If you walk 3 miles north and 4 miles east, you're √(3²+4²) = 5 miles from your start. Screen Size: TV and monitor sizes are measured diagonally. A screen that's 24 inches wide and 18 inches tall has a diagonal of √(24²+18²) = √(576+324) = √900 = 30 inches. Sports: Baseball diamond dimensions (90-foot baselines form a right triangle; the distance from home to second base is 90√2 ≈ 127.3 feet). Engineering: Calculating cable lengths, ramp slopes, roof pitches, and structural supports all rely on this ancient but essential theorem.
Frequently Asked Questions
What is the Pythagorean theorem and when do I use it?
The Pythagorean theorem states: a² + b² = c², where a and b are the two legs (shorter sides) of a right triangle, and c is the hypotenuse (longest side opposite the right angle). Use it ONLY for right triangles (triangles with one 90° angle). Example: If a ladder (hypotenuse) is 13 feet long and the base (leg a) is 5 feet from the wall, the top (leg b) is √(13² - 5²) = √(169 - 25) = √144 = 12 feet high.
How do I find the hypotenuse?
To find the hypotenuse (c), use the formula c = √(a² + b²). Square both legs, add them together, then take the square root. Example: If a = 3 and b = 4, then c = √(3² + 4²) = √(9 + 16) = √25 = 5. The hypotenuse is always the longest side in a right triangle.
How do I find a leg when I know the hypotenuse?
To find a leg when you know the hypotenuse (c) and the other leg, use: a = √(c² - b²) or b = √(c² - a²). Rearranging a² + b² = c² gives a² = c² - b². Example: If the hypotenuse is 10 and one leg is 6, then the other leg = √(10² - 6²) = √(100 - 36) = √64 = 8. Note: c must be larger than the known leg, or you get an invalid result.
What are Pythagorean triples?
Pythagorean triples are sets of three whole numbers (a, b, c) that satisfy a² + b² = c². The most common ones are: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), and (20, 21, 29). Any multiple of these also works: (6, 8, 10), (9, 12, 15), etc. These are useful for quickly solving right triangle problems with integer sides, like in carpentry or construction.
Does the Pythagorean theorem work for non-right triangles?
No! The Pythagorean theorem ONLY works for right triangles (triangles with exactly one 90° angle). For other triangles, use the Law of Cosines. You can use the 'Check Right Triangle' feature in this calculator to verify whether three sides form a right triangle—if they do, then a² + b² = c².
How accurate is this calculator?
This calculator uses JavaScript's built-in Math.sqrt() function with double-precision floating-point arithmetic (15-17 significant digits). Results are displayed to 4 decimal places. For practical purposes in construction, homework, or everyday calculations, this accuracy is more than sufficient.
Is this tool free to use?
Yes! Completely free with no hidden costs, subscriptions, or limitations. Solve as many right triangle problems as you need.