How the Pythagorean Theorem Works (With Real Examples)

The Pythagorean theorem is one of the most fundamental relationships in all of mathematics. It connects the three sides of a right triangle through a simple equation that has been known for over 4,000 years. From checking whether a wall is square to calculating distances on a map, this single formula solves an extraordinary range of practical problems.

This guide explains the theorem from the ground up, demonstrates multiple proofs, walks through real-world applications, and provides step-by-step examples you can follow with any right triangle.

What Is the Pythagorean Theorem?

The Pythagorean theorem states that in any right triangle, the square of the length of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the lengths of the other two sides (called legs). In its most recognizable form:

The theorem works in both directions. If three sides of a triangle satisfy a^2 + b^2 = c^2, the triangle is guaranteed to be a right triangle. This bidirectional property makes it equally useful for calculating unknown side lengths and verifying right angles.

While named after the Greek mathematician Pythagoras (circa 570-495 BCE), the relationship was known to Babylonian mathematicians as early as 1800 BCE, as evidenced by the Plimpton 322 clay tablet, and independently to mathematicians in ancient India and China.

The Formula Explained

The formula a^2 + b^2 = c^2 has three distinct uses depending on which side you need to find:

When solving for a leg, always subtract the smaller square from the larger square. The expression under the square root must be positive, which is guaranteed because the hypotenuse is always the longest side of a right triangle.

Geometric Proofs

The Pythagorean theorem has been proven in more than 370 distinct ways, making it one of the most proven results in mathematics. Two classic proofs illustrate the core idea.

Proof by rearrangement: Start with a large square with side length (a + b). Inside this square, arrange four identical right triangles with legs a and b. The triangles can be positioned in two different ways. In the first arrangement, the uncovered area in the center forms a single square with side length c, so the uncovered area equals c^2. In the second arrangement, the uncovered area forms two separate squares: one with side a (area a^2) and one with side b (area b^2). Since the total square area is the same and the four triangles are identical in both arrangements, the uncovered areas must be equal: a^2 + b^2 = c^2.

Proof by similar triangles: Draw the altitude from the right angle to the hypotenuse, dividing the original triangle into two smaller triangles. All three triangles (the original and the two smaller ones) are similar, meaning they have the same angles and proportional sides. Setting up the proportions between corresponding sides and simplifying algebraically yields a^2 + b^2 = c^2. This proof, essentially Euclid's approach from his Elements (Book I, Proposition 47), uses only basic geometry.

In 1876, James A. Garfield, who later became the 20th President of the United States, published a proof using a trapezoid constructed from two copies of a right triangle and a third isosceles right triangle. This creative approach demonstrates that new proofs continue to be discovered.

Pythagorean Triples

A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy a^2 + b^2 = c^2. These integer solutions create right triangles with clean, whole-number side lengths.

A Pythagorean triple is primitive if the three numbers share no common factor other than 1. Every primitive triple can be generated using the formula a = m^2 - n^2, b = 2mn, c = m^2 + n^2, where m and n are positive integers with m greater than n, m and n are not both odd, and they share no common factor. Any multiple of a Pythagorean triple is also a Pythagorean triple.

Step-by-Step Calculation Examples

Real-World Applications

The Pythagorean theorem appears in a wide range of practical fields:

• Construction and carpentry: Verifying square corners, calculating roof rafter lengths, determining staircase dimensions, and laying out building foundations all rely on the theorem. The 3-4-5 check is an everyday tool on construction sites worldwide.
• Navigation and mapping: Calculating the straight-line distance between two points on a grid uses the theorem directly. GPS systems use three-dimensional extensions of the same principle when computing distances between coordinates.
• Computer graphics: Distance calculations between pixels, collision detection in video games, and rendering three-dimensional scenes all apply the Pythagorean theorem thousands of times per second.
• Surveying: Land surveyors use the theorem to calculate distances across obstacles like rivers or ravines by setting up right triangles with measurable legs and computing the inaccessible hypotenuse.
• Electrical engineering: The relationship between resistance, reactance, and impedance in AC circuits follows the same a^2 + b^2 = c^2 relationship, where impedance is the hypotenuse of a right triangle formed by resistance and reactance.

Extensions and Generalizations

The Pythagorean theorem has been generalized in several important directions:

The distance formula in coordinate geometry is a direct application. The distance between points (x1, y1) and (x2, y2) is d = sqrt((x2 - x1)^2 + (y2 - y1)^2). This is the hypotenuse of a right triangle with legs (x2 - x1) and (y2 - y1).

The law of cosines generalizes the theorem to any triangle: c^2 = a^2 + b^2 - 2ab cos(C). When the angle C is 90 degrees, cos(90) = 0 and the formula reduces to the standard Pythagorean theorem.

Higher dimensions extend the concept naturally. In three dimensions, the diagonal of a rectangular box with sides a, b, and c is d = sqrt(a^2 + b^2 + c^2). This pattern continues for any number of dimensions.

Fermat's Last Theorem asks whether the equation a^n + b^n = c^n has positive integer solutions for n greater than 2. For n = 2, there are infinitely many solutions (Pythagorean triples). Andrew Wiles proved in 1995 that no solutions exist for any higher power, resolving a conjecture that had been open for 358 years.

Tips for Working with Right Triangles

• Always identify the hypotenuse first. The hypotenuse is opposite the right angle and is the longest side. Labeling it correctly prevents substitution errors in the formula.
• Memorize common Pythagorean triples. Recognizing that 3-4-5, 5-12-13, and 8-15-17 are right triangles saves calculation time and provides quick sanity checks for your results.
• Check your answer by squaring. After finding an unknown side, verify by substituting all three values back into a^2 + b^2 = c^2. If the equation balances, your calculation is correct.
• Use the theorem to verify right angles. Before assuming a corner is 90 degrees, measure all three sides and check the Pythagorean relationship. Physical measurements can be slightly off, so allow small tolerances.
• Simplify before computing. If both legs are multiples of the same number, factor it out first. For legs of 15 and 20, recognize this is 5 times the 3-4-5 triple, so the hypotenuse is 5 x 5 = 25 without any squaring or square roots.

Common Mistakes to Avoid

• Squaring the hypotenuse on the wrong side. Remember that c^2 equals the sum of a^2 and b^2, not the difference. When finding a leg, subtract: a = sqrt(c^2 - b^2). Accidentally adding when you should subtract gives an impossibly large result.
• Applying the theorem to non-right triangles. The formula a^2 + b^2 = c^2 only works when one angle is exactly 90 degrees. For other triangles, use the law of cosines or law of sines.
• Forgetting to take the square root. The theorem gives c^2, not c. A common error is stopping at a^2 + b^2 = 100 and reporting the answer as 100 instead of taking the square root to get 10.
• Using the slant height as the perpendicular height. In construction applications, the vertical rise and horizontal run must be measured perpendicular to each other. Measuring along a slope gives incorrect values.
• Rounding too early. When chaining calculations, keep full precision until the final step. Rounding intermediate square roots introduces compounding errors that grow with each subsequent calculation.

Last updated: February 23, 2026