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Pythagorean Theorem

Pythagorean Theorem

An equation relating the lengths of the sides of a right triangle. The sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse.

 

Right triangle with legs labeled a and b, hypotenuse labeled c, and the equation a² + b² = c²

 

 

 

See also

Theorem, Pythagorean triple, Pythagorean identities

Key Formula

a2+b2=c2a^2 + b^2 = c^2
Where:
  • aa = Length of one leg of the right triangle
  • bb = Length of the other leg of the right triangle
  • cc = Length of the hypotenuse (the side opposite the right angle)

Worked Example

Problem: A right triangle has legs of length 5 and 12. Find the length of the hypotenuse.
Step 1: Write the Pythagorean Theorem and identify the known values. Here, a = 5 and b = 12, and we need to find c.
a2+b2=c2a^2 + b^2 = c^2
Step 2: Substitute the known leg lengths into the equation.
52+122=c25^2 + 12^2 = c^2
Step 3: Compute the squares and add them together.
25+144=c2  ⟹  169=c225 + 144 = c^2 \implies 169 = c^2
Step 4: Take the positive square root of both sides to solve for c.
c=169=13c = \sqrt{169} = 13
Answer: The hypotenuse is 13 units long.

Another Example

This example solves for a missing leg rather than the hypotenuse, requiring you to rearrange the formula by subtracting instead of simply adding.

Problem: A right triangle has a hypotenuse of length 10 and one leg of length 6. Find the length of the other leg.
Step 1: Write the Pythagorean Theorem. This time you know the hypotenuse (c = 10) and one leg (a = 6), and you need to find the other leg b.
a2+b2=c2a^2 + b^2 = c^2
Step 2: Substitute the known values.
62+b2=1026^2 + b^2 = 10^2
Step 3: Compute the squares.
36+b2=10036 + b^2 = 100
Step 4: Isolate b² by subtracting 36 from both sides.
b2=100−36=64b^2 = 100 - 36 = 64
Step 5: Take the positive square root.
b=64=8b = \sqrt{64} = 8
Answer: The missing leg is 8 units long.

Frequently Asked Questions

Does the Pythagorean Theorem work for all triangles?
No. The Pythagorean Theorem applies only to right triangles — triangles that have exactly one 90° angle. For non-right triangles, you need the Law of Cosines, which is a generalization: c² = a² + b² − 2ab cos(C). When angle C is 90°, cos(90°) = 0 and the Law of Cosines reduces to the Pythagorean Theorem.
How can you use the Pythagorean Theorem to find distance between two points?
The distance formula on the coordinate plane is derived directly from the Pythagorean Theorem. Given two points (x₁, y₁) and (x₂, y₂), the horizontal distance |x₂ − x₁| and vertical distance |y₂ − y₁| form the legs of a right triangle. The distance between the points is the hypotenuse: d = √((x₂ − x₁)² + (y₂ − y₁)²).
What is a Pythagorean triple?
A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy a² + b² = c². Common examples include (3, 4, 5), (5, 12, 13), and (8, 15, 17). Any multiple of a Pythagorean triple is also a Pythagorean triple — for instance, (6, 8, 10) is simply (3, 4, 5) multiplied by 2.

Pythagorean Theorem vs. Law of Cosines

Pythagorean TheoremLaw of Cosines
Formulaa² + b² = c²c² = a² + b² − 2ab cos(C)
Applies toRight triangles onlyAny triangle
What you need to knowTwo side lengthsTwo sides and the included angle, or all three sides
RelationshipSpecial case of the Law of Cosines when C = 90°Generalization of the Pythagorean Theorem

Why It Matters

The Pythagorean Theorem appears throughout algebra, geometry, trigonometry, and physics. You use it every time you compute the distance between two points on a coordinate plane, find diagonal lengths in rectangles, or resolve vector components in physics. Standardized tests like the SAT, ACT, and state exams consistently include problems that require this theorem, making it one of the most essential formulas to memorize.

Common Mistakes

Mistake: Using a leg length in place of the hypotenuse (assigning c to one of the shorter sides).
Correction: The variable c must always represent the hypotenuse — the longest side, directly opposite the 90° angle. The two legs are a and b, and it does not matter which leg you call a or b since addition is commutative.
Mistake: Adding the side lengths instead of their squares, e.g., writing a + b = c.
Correction: The theorem requires you to square each side first: a² + b² = c². For a 3-4-5 triangle, 3 + 4 ≠ 5, but 9 + 16 = 25 ✓. Always square before you add.

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