Pythagorean Theorem Calculator

The Pythagorean theorem is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides (legs).

$a^2 + b^2 = c^2$

where:

• $a$ and $b$ are the lengths of the two legs
• $c$ is the length of the hypotenuse (the longest side)

Solving for Each Side

• Hypotenuse: $c = \sqrt{a^2 + b^2}$
• Leg $a$: $a = \sqrt{c^2 - b^2}$
• Leg $b$: $b = \sqrt{c^2 - a^2}$

Historical Note

Named after the ancient Greek mathematician Pythagoras (c. 570–495 BC), this theorem was known to Babylonian mathematicians over a thousand years earlier. It is one of the most proven theorems in mathematics, with hundreds of distinct proofs.

Pythagorean Triples

A Pythagorean triple consists of three positive integers $a$, $b$, $c$ that satisfy $a^2 + b^2 = c^2$. Common examples:

• $(3, 4, 5)$
• $(5, 12, 13)$
• $(8, 15, 17)$
• $(7, 24, 25)$

Step-by-Step Process

1. Identify the right angle and label the sides: $a$, $b$ (legs) and $c$ (hypotenuse)
2. Determine which side is unknown
3. Substitute the known values into $a^2 + b^2 = c^2$
4. Solve for the unknown side
5. Simplify the result (exact or decimal form)

Finding the Hypotenuse

Given legs $a$ and $b$:

$c = \sqrt{a^2 + b^2}$

Example: If $a = 6$ and $b = 8$, then $c = \sqrt{36 + 64} = \sqrt{100} = 10$.

Finding a Leg

Given hypotenuse $c$ and one leg $a$:

$b = \sqrt{c^2 - a^2}$

Example: If $c = 13$ and $a = 5$, then $b = \sqrt{169 - 25} = \sqrt{144} = 12$.

Checking if a Triangle is Right

Given three sides, check if $a^2 + b^2 = c^2$ (where $c$ is the longest side):

• If $a^2 + b^2 = c^2$: right triangle
• If $a^2 + b^2 > c^2$: acute triangle
• If $a^2 + b^2 < c^2$: obtuse triangle

Distance Formula Connection

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is derived from the Pythagorean theorem:

$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

Common Formulas

• Confusing the hypotenuse with a leg — the hypotenuse is always the longest side opposite the right angle. Using it as a leg in the formula gives wrong results.
• Forgetting to take the square root — after computing $a^2 + b^2$, you must take $\sqrt{a^2 + b^2}$ to get $c$, not leave it as $a^2 + b^2$.
• Subtracting in the wrong direction — when finding a leg, compute $c^2 - a^2$, not $a^2 - c^2$ (which would give a negative number under the radical).
• Applying the theorem to non-right triangles — the Pythagorean theorem only works for right triangles. For other triangles, use the Law of Cosines.
• Rounding too early — keep the exact value under the square root as long as possible to maintain accuracy.