Pythagorean Theorem — Formula and Exercises

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs. With legs $a$ and $b$ and hypotenuse $c$ :

a^2 + b^2 = c^2

The symbols matter. The legs $a$ and $b$ are the two sides that form the $90^\circ$ angle; the hypotenuse $c$ is the side opposite that angle, and it is always the longest side. You can only use the formula when an angle of $90^\circ$ is present.

Why it is about squares, not sides

The key relationship is not $a + b = c$ , but a relationship between squares. The theorem compares the areas of squares built on each side, which is exactly why $a^2$ , $b^2$ , and $c^2$ appear instead of the plain side lengths. Holding onto that idea keeps you from the tempting but wrong shortcut of just adding the two shorter sides.

Naming the sides correctly is the other half of the setup. Legs form the right angle; the hypotenuse sits opposite it. Put another side in place of $c$ and the calculation can look tidy yet be wrong from the very first line.

Worked example: finding the hypotenuse

Take a right triangle with legs of $6$ cm and $8$ cm and find the hypotenuse. Set up the formula:

6^2 + 8^2 = c^2

Compute the squares:

36 + 64 = c^2

Add:

100 = c^2

Take the positive square root:

c = 10

The hypotenuse measures $10$ cm. This passes a quick sanity check: it must be longer than both $6$ cm and $8$ cm, and it is.

Try the calculation yourself

Practice 1. Solve a right triangle with legs $5$ and $12$ . Applied correctly, the hypotenuse comes out to $13$ .

Practice 2 — finding a leg. If you know $c$ and $b$ , do not add two squares. Subtract:

a^2 = c^2 - b^2

This is also how the theorem powers distance on a grid: moving $3$ units across and $4$ up, the direct distance is

\sqrt{3^2 + 4^2} = 5

the same idea that later becomes the distance formula between two points.

Calculation traps to avoid

Wrong triangle. Using the theorem when there is no right angle. The relationship generally fails without one — check for the $90^\circ$ angle first.
Leg vs. hypotenuse. Treating a leg as $c$ . The hypotenuse is the side opposite the right angle.
Adding when you should subtract. To find a leg from the hypotenuse, subtract squares; do not add them.
Stopping too early. Reaching $c^2 = 100$ is not the answer. The length is $c = 10$ , not $100$ — take the square root.

Before plugging in numbers, confirm two things: there is a right angle, and you have correctly identified the hypotenuse. With both met, the theorem is the right tool.

Frequently Asked Questions

When can you use the Pythagorean theorem?: Only in a right triangle, meaning a triangle with a 90 degree angle. The theorem says the square of the hypotenuse equals the sum of the squares of the legs. If the triangle is not a right triangle, the relationship generally does not hold, so checking for the right angle is the first thing to do before plugging in numbers.
How do you find the hypotenuse of a right triangle?: Square each leg, add the results, and take the positive square root. For example, with legs of 6 cm and 8 cm, you get 36 plus 64 equals 100, so the hypotenuse is 10 cm. A quick sanity check: the hypotenuse must always come out longer than each leg.
How do you find a leg when you know the hypotenuse?: Do not add the two known squares. Instead, subtract: the unknown leg squared equals the hypotenuse squared minus the known leg squared. For instance, knowing c and b, compute a squared as c squared minus b squared, then take the square root. Adding instead of subtracting is one of the most common setup errors.
How do you identify the hypotenuse and the legs?: The legs are the two sides that form the right angle, and the hypotenuse is the side opposite the right angle, which is also the longest side. Naming the sides correctly before calculating prevents almost all setup errors, because putting another side in place of the hypotenuse makes the equation wrong from the start.
Why is the answer 10 and not 100 when c squared equals 100?: Because the theorem gives you the square of the side, not the side itself. Reaching c squared equals 100 is not the final step; you still need the positive square root, which gives c equals 10. Stopping too early and reporting the squared value is a common slip-up in Pythagorean theorem exercises.

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