The Pythagoras theorem states that in any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides — written algebraically as $a^2 + b^2 = c^2$, where $c$ is the hypotenuse and $a$ and $b$ are the two legs. The theorem holds for every right triangle, has at least $370$ distinct proofs, and shows up in everything from carpentry to GPS calculation. The Greeks formalised it; the Babylonians, Egyptians, Chinese, and Indians had been using it for centuries before Pythagoras was born.
Quick Reference
Field | Value |
|---|---|
Definition | In a right triangle, $a^2 + b^2 = c^2$ |
Symbol | $a, b, c$ for the sides; $c$ is the hypotenuse |
Formula | $c = \sqrt{a^2 + b^2}$ |
Type | Geometric theorem |
Used in | Geometry, trigonometry, distance formula, GPS, engineering |
What is the Pythagoras Theorem?
In any right-angled triangle, the side opposite the right angle is the longest — call it the hypotenuse. The Pythagoras theorem states that if you square the lengths of the two shorter sides and add them, the sum equals the square of the hypotenuse.
So if a right triangle has legs of length $3$ and $4$, the hypotenuse is $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$. The triple $(3, 4, 5)$ is the most famous example of a Pythagorean triple — three whole numbers that satisfy the theorem. Other common triples are $(5, 12, 13)$, $(8, 15, 17)$, $(7, 24, 25)$.
The converse is also true: if three sides $a, b, c$ of a triangle satisfy $a^2 + b^2 = c^2$, the triangle is necessarily right-angled at the corner opposite $c$. Builders use this fact daily — the $3$–$4$–$5$ rule for setting square corners is the converse in action.
Four Ways To Prove The Pythagoras Theorem
The theorem is the most-proved result in mathematics. Elisha Loomis's 1940 book The Pythagorean Proposition catalogued $370$ different proofs. Here are four that capture different angles.
Proof 1: Visual rearrangement. Take four identical right triangles with legs $a, b$ and hypotenuse $c$. Arrange them inside a square of side $a + b$ in two ways.
The first arrangement leaves an empty square of side $c$ in the middle. The second arrangement leaves two empty squares of sides $a$ and $b$. Both empty regions have the same total area, because the four triangles take up the same area both times. Therefore $c^2 = a^2 + b^2$.
Proof 2 (algebraic, using the same arrangement). The big square has area $(a+b)^2 = a^2 + 2ab + b^2$. The four triangles have total area $4 \cdot \frac{1}{2} ab = 2ab$. The middle square has area $c^2$.
So $c^2 + 2ab = a^2 + 2ab + b^2$, which simplifies to $a^2 + b^2 = c^2$.
Proof 3 (similar triangles). Drop a perpendicular from the right-angle vertex to the hypotenuse. This splits the original triangle into two smaller triangles, both similar to the original. Comparing ratios from those similarities gives the result directly.
Proof 4 (Garfield's proof). US President James A. Garfield, before he became president, published a proof using a trapezium of two right triangles. The trapezium's area, computed two ways, again yields $a^2 + b^2 = c^2$.
Garfield was a mathematics professor before politics; the proof appeared in the New England Journal of Education in $1876$.
Worked Examples of Pythagoras Theorem
Example 1: Find the hypotenuse
A ladder leans against a wall. The foot of the ladder is $5,\text{m}$ from the wall, and the top reaches $12,\text{m}$ up. How long is the ladder?
The ladder forms the hypotenuse of a right triangle with legs $5$ and $12$.
$$c^2 = 5^2 + 12^2 = 25 + 144 = 169$$
$$c = \sqrt{169} = 13$$
Final answer: The ladder is $13,\text{m}$ long.
Example 2: Find a missing leg (the wrong path first)
A right triangle has hypotenuse $25$ and one leg $7$. Find the other leg.
The instinct is to write $a^2 + b^2 = c^2$ as $7^2 + b^2 = 25^2$ and grind through. Try the wrong path first, where many students go: set up $25^2 + 7^2 = b^2$ instead, treating the longest given side as a leg.
$$b^2 = 625 + 49 = 674, \quad b = \sqrt{674} \approx 25.96$$
That cannot be right — $b$ would be longer than the hypotenuse, which violates the geometry. Stop and re-check.
The correct setup keeps $25$ as the hypotenuse:
$$7^2 + b^2 = 25^2$$
$$49 + b^2 = 625$$
$$b^2 = 576$$
$$b = 24$$
Final answer: The other leg is $24$.
Example 3: Use the converse to check a right angle
A triangle has sides $9, 40, 41$. Is it right-angled?
Check whether $9^2 + 40^2 = 41^2$:
$$9^2 + 40^2 = 81 + 1600 = 1681$$
$$41^2 = 1681$$
The two are equal, so the triangle is right-angled at the corner opposite the side of length $41$.
Final answer: Yes — $(9, 40, 41)$ is a Pythagorean triple, so the triangle is right-angled.
The Mathematicians Who Shaped The Pythagoras Theorem
The story does not begin with Pythagoras. The Babylonian tablet YBC 7289 (around $1800$ BCE) shows a unit square with its diagonal labelled in cuneiform numerals — and the diagonal value, in modern decimal, is $1.41421296$, accurate to six decimal places of $\sqrt{2}$. Babylonian scribes were calculating with the theorem more than a thousand years before the Greek school.
Pythagoras of Samos (c. $570$–$495$ BCE, Greece) led a philosophical-mathematical school that proved the result with the geometric arguments we still teach. The school treated numbers as quasi-mystical, and legend has it that one member, Hippasus, was killed for revealing that $\sqrt{2}$ was irrational — a discovery the school had tried to suppress because it broke their philosophy that all quantities could be expressed as ratios.
Bhāskara II (1114–1185, India) gave a one-word proof, drawing the rearrangement diagram and writing simply "Behold!" — the visual proof speaks for itself. Euclid (c. $300$ BCE, Greece) gave the most cited rigorous proof, in Elements Book I, Proposition $47$.
Common Mistakes of Pythagoras Theorem
Mistake 1: Treating the longest given side as a leg.
Where it slips in: When the problem gives one leg and the hypotenuse, but the student hasn't paused to identify which is which.
Don't do this: $25^2 + 7^2 = b^2$, getting $b \approx 25.96$, longer than the hypotenuse — geometrically impossible.
The correct way: Identify the hypotenuse first (always opposite the right angle, always the longest side). Then write $a^2 + b^2 = c^2$ with the hypotenuse on the right.
Mistake 2: Forgetting to take the square root.
Where it slips in: Computing $a^2 + b^2$, getting (say) $169$, and writing the answer as $169$ instead of $\sqrt{169} = 13$.
Don't do this: "$c^2 = 169$, so $c = 169$." Stop at the squared form.
The correct way: Compute $a^2 + b^2$, then take the square root. The rusher who skips this loses the entire question.
Mistake 3: Using the theorem on a non-right triangle.
Where it slips in: The problem doesn't explicitly say "right triangle" but the student assumes it is anyway.
Don't do this: Apply $a^2 + b^2 = c^2$ to any triangle.
The correct way: Confirm the triangle is right-angled — by inspection, by the right-angle mark, or by the converse test. For non-right triangles, use the Law of Cosines instead: $c^2 = a^2 + b^2 - 2ab\cos C$.
Mistake 4: Forgetting Pythagorean triples are not the only right triangles.
Where it slips in: The student memorises $(3, 4, 5)$, $(5, 12, 13)$, and tries to spot one in every problem.
Don't do this: Assume the answer must be a whole number.
The correct way: Most right triangles have irrational hypotenuse. A triangle with legs $1$ and $1$ has hypotenuse $\sqrt{2}$ — exactly the length that broke the Pythagorean school.
The real-world version of Mistake 1 is one of the oldest engineering errors documented in surveying. In ancient Egyptian rope-stretching for the pyramids, surveyors used the $3$–$4$–$5$ rule to set right angles. A misidentified hypotenuse meant a structure built off-square — and several pyramid-base errors visible to archaeologists today are traced to exactly this mistake. See Egyptian rope-stretchers (harpedonaptae) for the historical record.
Try These Next
Now try this: a right triangle has legs $9$ and $12$. Find the hypotenuse. Then check your answer against the $(3, 4, 5)$ family — what's the connection? If you get stuck, come back to Example $1$ and follow the same setup.
If your child is comfortable with the basic theorem, the natural next step is the distance formula (which is the Pythagoras theorem applied to coordinate-plane points) and then trigonometric ratios (which extend the theorem into angles). At Bhanzu, trainers connect each step in this chain so the formulas don't sit isolated in different chapters.
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