Pythagorean Theorem — Formula, Proof, and Examples

Grade: 8-9 | Topic: Geometry

What You Will Learn

By the end of this guide you will understand what the Pythagorean theorem says, why it is true, and how to use it. You will be able to find a missing side of any right triangle, check whether a triangle is a right triangle, and apply the theorem to real-world distance problems.

Theory

The theorem and its formula

The Pythagorean theorem is one of the most important results in all of mathematics. It describes the relationship between the three sides of a right triangle — a triangle that contains exactly one 90-degree angle.

In a right triangle the longest side, opposite the right angle, is called the hypotenuse ($c$). The other two sides are called the legs ($a$ and $b$). The theorem states:

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

In words: the sum of the squares of the legs equals the square of the hypotenuse.

For example, a right triangle with legs $a = 3$ and $b = 4$ has a hypotenuse of:

$c = \sqrt{a^2 + b^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

This is the famous 3-4-5 right triangle, the simplest example of a Pythagorean triple.

A visual proof (rearrangement proof)

Imagine a large square with side length $(a + b)$. You can fill this square two different ways:

Way 1 — four triangles and a small square: Place four copies of the right triangle (each with legs $a$ and $b$ and hypotenuse $c$) inside the big square so that they form a tilted square in the center. The area of the big square is $(a + b)^2$. The four triangles together have area $4 \times \frac{1}{2}ab = 2ab$, and the inner square has area $c^2$. So:

$(a + b)^2 = 2ab + c^2$

Way 2 — expand the left side:

$(a + b)^2 = a^2 + 2ab + b^2$

Setting the two expressions equal:

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

Subtract $2ab$ from both sides:

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

This elegant proof shows that the theorem is not just a formula to memorize — it is a geometric fact about how areas relate in a right triangle.

Finding a missing side

The formula $a^2 + b^2 = c^2$ can be rearranged depending on which side is unknown:

Finding the hypotenuse (when you know both legs):

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

Finding a leg (when you know the hypotenuse and one leg):

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

Always remember: the hypotenuse $c$ is the longest side. If you solve for a side and get a value larger than the hypotenuse, you have mixed up which side is which.

Checking if a triangle is a right triangle

Given three side lengths, you can verify whether they form a right triangle by testing the Pythagorean equation. Label the longest side as $c$ and check:

• If $a^2 + b^2 = c^2$, the triangle is a right triangle.
• If $a^2 + b^2 > c^2$, the triangle is acute (all angles less than 90 degrees).
• If $a^2 + b^2 < c^2$, the triangle is obtuse (one angle greater than 90 degrees).

Worked Examples

Example 1: Finding the hypotenuse (easy)

Problem: A right triangle has legs of length 6 cm and 8 cm. Find the hypotenuse.

Step 1: Write the Pythagorean theorem. $a^2 + b^2 = c^2$

Step 2: Substitute the known values. $6^2 + 8^2 = c^2$ $36 + 64 = c^2$ $100 = c^2$

Step 3: Take the square root of both sides. $c = \sqrt{100} = 10 \text{ cm}$

Answer: The hypotenuse is 10 cm.

Example 2: Finding a missing leg (medium)

Problem: A right triangle has a hypotenuse of 13 m and one leg of 5 m. Find the other leg.

Step 1: Write the formula solved for the unknown leg. $a = \sqrt{c^2 - b^2}$

Step 2: Substitute the known values. $a = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144}$

Step 3: Simplify. $a = 12 \text{ m}$

Answer: The missing leg is 12 m.

Example 3: Checking if a triangle is a right triangle (medium)

Problem: A triangle has sides of 7, 24, and 25. Is it a right triangle?

Step 1: Identify the longest side as the potential hypotenuse. $c = 25$, $a = 7$, $b = 24$.

Step 2: Check whether $a^2 + b^2 = c^2$. $7^2 + 24^2 = 49 + 576 = 625$ $25^2 = 625$

Step 3: Compare. $625 = 625 \checkmark$

Answer: Yes, this is a right triangle because $a^2 + b^2 = c^2$.

Example 4: Real-world ladder problem (challenging)

Problem: A 10-foot ladder leans against a wall. The base of the ladder is 6 feet from the wall. How high up the wall does the ladder reach?

Step 1: Visualize the right triangle. The ladder is the hypotenuse ($c = 10$ ft), the distance from the wall is one leg ($b = 6$ ft), and the height up the wall is the unknown leg ($a$).

Step 2: Apply the formula. $a = \sqrt{c^2 - b^2} = \sqrt{10^2 - 6^2} = \sqrt{100 - 36} = \sqrt{64}$

Step 3: Simplify. $a = 8 \text{ ft}$

Answer: The ladder reaches 8 feet up the wall.

Example 5: Distance between two points (challenging)

Problem: Find the distance between the points $A(1, 2)$ and $B(4, 6)$ on a coordinate plane.

Step 1: The horizontal distance is the difference in $x$-coordinates, and the vertical distance is the difference in $y$-coordinates. These form the two legs of a right triangle. $\Delta x = 4 - 1 = 3$ $\Delta y = 6 - 2 = 4$

Step 2: Apply the Pythagorean theorem (this is how the distance formula is derived). $d = \sqrt{(\Delta x)^2 + (\Delta y)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25}$

Step 3: Simplify. $d = 5$

Answer: The distance between $A$ and $B$ is 5 units.

Common Mistakes

Mistake 1: Forgetting to take the square root as the final step

❌ $c = 6^2 + 8^2 = 36 + 64 = 100$ (stopping here and claiming $c = 100$)

✅ $c = \sqrt{6^2 + 8^2} = \sqrt{100} = 10$

Why this matters: The formula gives you $c^2$, not $c$. You must take the square root to get the actual side length. Forgetting this step gives an answer that is far too large and has incorrect units (square units instead of linear units).

Mistake 2: Using the formula on non-right triangles

❌ A triangle with sides 5, 7, and 9: assuming $5^2 + 7^2 = 9^2$ and "solving" for a missing angle.

✅ First check: $25 + 49 = 74$, but $9^2 = 81$. Since $74 \neq 81$, this is not a right triangle. The Pythagorean theorem does not apply directly.

Why this matters: The theorem only works for right triangles. Applying it to other triangles gives incorrect results. Always confirm the triangle has a 90-degree angle before using $a^2 + b^2 = c^2$.

Mistake 3: Subtracting when you should add (or vice versa)

❌ Finding the hypotenuse: $c = \sqrt{13^2 - 5^2}$ (subtracting instead of adding)

✅ Finding the hypotenuse: $c = \sqrt{a^2 + b^2}$. Only subtract when finding a leg: $a = \sqrt{c^2 - b^2}$.

Why this matters: You add the squares of the two legs to find the hypotenuse, and you subtract a leg's square from the hypotenuse's square to find the other leg. Mixing these up gives a wrong (and sometimes impossible) answer.

Practice Problems

Try these on your own before checking the answers:

1. A right triangle has legs of 9 cm and 12 cm. Find the hypotenuse.
2. A right triangle has a hypotenuse of 17 m and one leg of 8 m. Find the other leg.
3. Do the sides 11, 60, and 61 form a right triangle?
4. A rectangular TV screen is 48 inches wide and 36 inches tall. What is the diagonal measurement (screen size)?
5. Find the distance between points $P(-2, 3)$ and $Q(4, -5)$.

Click to see answers

1. $c = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15$ cm.
2. $a = \sqrt{17^2 - 8^2} = \sqrt{289 - 64} = \sqrt{225} = 15$ m.
3. Check: $11^2 + 60^2 = 121 + 3600 = 3721$. And $61^2 = 3721$. Yes, it is a right triangle.
4. $d = \sqrt{48^2 + 36^2} = \sqrt{2304 + 1296} = \sqrt{3600} = 60$ inches.
5. $d = \sqrt{(4-(-2))^2 + (-5-3)^2} = \sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ units.

Summary

• The Pythagorean theorem states that in a right triangle, $a^2 + b^2 = c^2$, where $c$ is the hypotenuse and $a$, $b$ are the legs.
• To find the hypotenuse, add the squares of the legs and take the square root: $c = \sqrt{a^2 + b^2}$.
• To find a missing leg, subtract and take the square root: $a = \sqrt{c^2 - b^2}$.
• You can test whether three side lengths form a right triangle by checking if $a^2 + b^2 = c^2$.
• The theorem is the foundation for the distance formula used in coordinate geometry.

Related Topics

• Pythagorean Theorem Examples with Step-by-Step Solutions
• Pythagorean Theorem Word Problems with Answers
• Distance Formula — How to Find Distance Between Two Points
• Pythagorean Triples — List, Formula, and How to Find Them

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