Pythagorean Theorem Calculator — Free Online Tool

How to Use the Pythagorean Theorem Calculator

1. Choose what to solve for: Use the radio buttons to select which side of the right triangle you want to find. Select "Solve for c (hypotenuse)" to find the longest side, or "Solve for a" or "Solve for b" to find one of the shorter legs. The input fields automatically adjust based on your selection.
2. Enter the known values: Input the two side lengths you already know. When solving for the hypotenuse, enter both legs (a and b). When solving for a leg, enter the other leg and the hypotenuse. All values must be positive numbers. The hypotenuse must be longer than either individual leg for the calculation to work.
3. Review the results: The calculator instantly shows the missing side value, a detailed step-by-step computation showing each mathematical operation, all three side lengths displayed together, and a verification section that confirms a² + b² equals c². Results update in real time as you modify inputs.

The default values are the classic 3-4-5 Pythagorean triple, which you can see verified in the results panel. Change the values to solve any right triangle problem you encounter.

Pythagorean Theorem Formulas

Variables Explained

• a (leg): One of the two shorter sides of the right triangle that form the right angle. In the standard orientation, this is typically the vertical side.
• b (leg): The other shorter side of the right triangle forming the right angle. In the standard orientation, this is typically the horizontal side. The labels a and b are interchangeable.
• c (hypotenuse): The longest side of the right triangle, located opposite the 90-degree angle. The hypotenuse is always longer than either leg individually but shorter than the sum of both legs.

Step-by-Step Example

Find the hypotenuse of a right triangle with legs a = 5 and b = 12:

1. Write the theorem: a² + b² = c²
2. Substitute known values: 5² + 12² = c²
3. Calculate squares: 25 + 144 = c²
4. Add: 169 = c²
5. Take the square root: c = √169 = 13

The hypotenuse is 13 units. The set (5, 12, 13) is a well-known Pythagorean triple, meaning all three values are whole numbers that perfectly satisfy the theorem.

Practical Examples

Example 1: Tom's Ladder Placement

Tom needs to reach a window 15 feet above the ground. Safety guidelines recommend placing the ladder base 5 feet from the wall. He needs to determine the minimum ladder length required.

• Wall height (a) = 15 feet, base distance (b) = 5 feet
• c = √(15² + 5²) = √(225 + 25) = √250 ≈ 15.81 feet
• Tom needs at least a 16-foot ladder

A 16-foot ladder will reach the 15-foot window with the base 5 feet from the wall. The actual ladder extends slightly beyond the window, which is recommended for safe climbing. Tom should also ensure the ladder angle is between 70 and 80 degrees for stability.

Example 2: Sarah's TV Size Verification

Sarah bought a 55-inch TV and wants to verify the stated size. She measures the screen width as 47.9 inches and the height as 27 inches. TV sizes are measured diagonally, so she uses the Pythagorean theorem.

• Width (a) = 47.9 inches, Height (b) = 27 inches
• Diagonal = √(47.9² + 27²) = √(2,294.41 + 729) = √3,023.41 ≈ 54.99 inches
• The diagonal is approximately 55 inches, confirming the advertised size

The measured diagonal of 54.99 inches confirms the TV is indeed a 55-inch model. This method works for any rectangular screen, monitor, or display. The aspect ratio of modern TVs is 16:9, meaning the width-to-height ratio is always 16 to 9.

Example 3: James's Property Diagonal

James owns a rectangular lot that is 120 feet wide and 160 feet deep. He wants to install a diagonal walkway across the property and needs to know the exact length.

• Width (a) = 120 feet, Depth (b) = 160 feet
• Diagonal = √(120² + 160²) = √(14,400 + 25,600) = √40,000 = 200 feet
• Walkway materials needed: 200 feet of pavers (3 feet wide = 600 sq ft of pavers)

The diagonal walkway will be exactly 200 feet long. Interestingly, 120-160-200 is a multiple of the 3-4-5 Pythagorean triple (multiplied by 40). For area calculations for the walkway materials, try our area calculator.

Example 4: Elena's Roof Rafter Calculation

Elena is building a shed with a roof that rises 4 feet over a horizontal run of 6 feet. She needs to calculate the rafter length to cut the lumber correctly.

• Rise (a) = 4 feet, Run (b) = 6 feet
• Rafter = √(4² + 6²) = √(16 + 36) = √52 ≈ 7.21 feet
• Adding 1 foot for overhang: 7.21 + 1 = 8.21 feet per rafter

Each rafter needs to be approximately 8.25 feet (8 feet 3 inches) to account for the calculated length plus overhang. Elena should cut the rafters from 10-foot lumber to leave room for angle cuts at both ends. The roof pitch is 4/6 or about 33.7 degrees, which provides good water drainage.

Pythagorean Triples Reference Table

Tips and Complete Guide

History of the Pythagorean Theorem

While named after the Greek mathematician Pythagoras (c. 570-495 BCE), evidence of this theorem predates him by over a millennium. A Babylonian clay tablet known as Plimpton 322 (dating to about 1800 BCE) contains a list of Pythagorean triples, suggesting the Babylonians understood and applied the relationship. Ancient Indian mathematicians documented the theorem in the Sulba Sutras around 800 BCE. Chinese mathematicians proved it independently in the Zhoubi Suanjing around 1000 BCE. Despite its ancient origins, the theorem remains one of the most frequently used mathematical tools in modern engineering, physics, and computer science.

Applications in Coordinate Geometry

The Pythagorean theorem is the foundation of the distance formula in coordinate geometry. To find the distance between two points (x₁, y₁) and (x₂, y₂), you calculate d = √((x₂-x₁)² + (y₂-y₁)²), which is the Pythagorean theorem applied to the horizontal and vertical differences. This extends to three dimensions as d = √((x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²). Every GPS system, mapping application, and game engine uses this formula to calculate distances. Our distance calculator applies this formula for both 2D and 3D coordinates.

The 3-4-5 Method for Checking Right Angles

The most practical application of the Pythagorean theorem in construction is the 3-4-5 method. To check if a corner is a perfect right angle, measure 3 feet along one wall from the corner, 4 feet along the other wall, and then measure the diagonal between those two points. If the diagonal is exactly 5 feet, the corner is square. For larger projects, use multiples: 6-8-10, 9-12-15, or 12-16-20. Larger measurements give more accurate results because small measurement errors become proportionally smaller. Professional builders frequently use this method when laying foundations, framing walls, and installing tile.

Special Right Triangles

Two special right triangles appear frequently in mathematics and are worth memorizing. The 45-45-90 triangle has legs in a 1:1 ratio and a hypotenuse of √2 times the leg length. The 30-60-90 triangle has sides in a 1:√3:2 ratio. These ratios are derived directly from the Pythagorean theorem and trigonometry. The 45-45-90 triangle describes the diagonal of a square, while the 30-60-90 triangle describes half of an equilateral triangle. Recognizing these patterns can significantly speed up calculations in geometry, trigonometry, and standardized tests.

Common Mistakes to Avoid

• Applying to non-right triangles: The theorem a² + b² = c² works only for right triangles. Using it on other triangles gives incorrect results. For non-right triangles, use the law of cosines instead.
• Confusing legs with hypotenuse: The hypotenuse (c) is always the longest side and must be opposite the right angle. Labeling the wrong side as c leads to impossible results like taking the square root of a negative number.
• Forgetting to take the square root: After calculating a² + b², remember that c² = a² + b², not c = a² + b². You must take the square root of the sum to find c.
• Rounding too early: When solving multi-step problems, carry extra decimal places through intermediate calculations and round only the final answer. Early rounding accumulates error.
• Assuming integer results: While Pythagorean triples produce whole numbers, most real-world problems yield irrational numbers. Legs of 4 and 7 give c = √65 ≈ 8.062, not a whole number.

Frequently Asked Questions