It might seem like Pythagorean Theorem is all about right triangles, but if we dig deeper it's also about squares.

That's right! Let’s say we have a right triangle with sides $a$, $b$, and $c$.

$a$

$b$

$c$

In a right triangle, the mini square 🔲 marks the right angle. Sides $a$ and $b$ form an L around the right angle and are called the legs.

The small square inside the triangle always points towards $c$, the longest side of the triangle, also called the hypotenuse.

The Pythagorean Theorem tells us that if squares are formed on each of the three sides, the sum of the areas of the two smaller squares is equal to the area of the biggest square.

$a$

$b$

$c$

$a_{2}+b_{2}=c_{2}$

We can use this concept to find any missing side of a right triangle given the two other sides.

Check out our step-by-step Pythagorean Theorem calculator or continue the lesson to see this in practice!

Pythagorean Theorem Calculator

Step 1. Identify the given and missing sides.

The small square inside the triangle points toward the hypotenuse (the longest side). The legs are the sides that form the right angle and L shape of the triangle.

$a$

$b$

$c$

Given Values:Missing:

Please enter your given values.

Step 3. Use Pythagorean's Theorem to find the missing side.

Pythagorean's Theorem tells us the areas of the smaller two squarescombined are equal to the area of the largest square. We can use this to find the missing side.

$a_{2}+b_{2}0_{2}+0_{2}0+000 0 =c_{2}=c_{2}=c_{2}=c_{2}=c=c $Nice! The hypotenuse of the triangle is $0$.

Key Steps 🗝 How to Use the Pythagorean Theorem

Step 1. Identify the given and missing sides.

The small square inside the triangle points toward the hypotenuse (the longest side). The legs are the sides that form the right angle and L shape of the triangle.

$a$

$b$

$c$

Step 2. Imagine a square on each of the three sides of the triangle.

$a$

$b$

$c$

Step 3. Use Pythagorean's Theorem to find the missing side.

Pythagorean's Theorem tells us the areas of the smaller two squarescombined are equal to the area of the largest square. We can use this to find the missing side:

$a_{2}+b_{2}=c_{2}$

Continue to explore some problems using Pythagorean's Theorem!

You can also use the Quick Links menu on the left to jump to a section of your choice.

You can also use the Quick Links dropdown above to jump to a section of your choice.

Pythagorean Theorem

The Pythagorean Theorem helps us solve for missing sides in right triangles.$a_{2}+b_{2}=c_{2}$One way to remember this formula is to imagine forming squares on each of the three sides of a right triangle. The areas of the smaller two squares combined are equal to the area of the largest square.

$a$

$b$

$c$

The smaller squares are formed by the legs of the triangle, which are referred to as $a$ and $b$ in the Pythagorean Theorem. The biggest square is formed by the hypotenuse, which is referred to as $c$ in the Pythagorean Theorem.

Let’s review some example problems to see how we can use the Pythagorean Theorem to find different missing sides of right triangles.

Let's start by finding the length of the missing side in this triangle:

$?$

$4$

$5$

Step 1. Identify the given and missing sides.

The small square inside the triangle points toward the hypotenuse (the longest side). The legs are the sides that form the right angle and L shape of the triangle.

$?$

$4$

$5$

Step 3. Use Pythagorean's Theorem to find the missing side.

Pythagorean's Theorem tells us the areas of the smaller two squarescombined are equal to the area of the largest square. We can use this to find the missing side.

$a_{2}+b_{2}a_{2}+4_{2}a_{2}+16a_{2}a_{2}a =c_{2}=5_{2}=25=25−16=9=9 =3 $Nice! The length of the missing side of the triangle is $3$.

Amazing! Now, let's find the the missing side in this triangle:

$8$

$?$

$10$

Step 1. Identify the given and missing sides.

The small square inside the triangle points toward the hypotenuse (the longest side). The legs are the sides that form the right angle and L shape of the triangle.

$8$

$?$

$10$

Step 3. Use Pythagorean's Theorem to find the missing side.

Pythagorean's Theorem tells us the areas of the smaller two squarescombined are equal to the area of the largest square. We can use this to find the missing side.

$a_{2}+b_{2}8_{2}+b_{2}64+b_{2}b_{2}b_{2}b =c_{2}=10_{2}=100=100−64=36=36 =6 $Nice! The length of the missing side of the triangle is $6$.

Incredible! Try one last problem. Find the missing side in this triangle:

$5$

$12$

$?$

Step 1. Identify the given and missing sides.

The small square inside the triangle points toward the hypotenuse (the longest side). The legs are the sides that form the right angle and L shape of the triangle.

$5$

$12$

$?$

Step 3. Use Pythagorean's Theorem to find the missing side.

Pythagorean's Theorem tells us the areas of the smaller two squarescombined are equal to the area of the largest square. We can use this to find the missing side.

$a_{2}+b_{2}5_{2}+12_{2}25+144169169 13 =c_{2}=c_{2}=c_{2}=c_{2}=c=c $Nice! The hypotenuse of the triangle is $13$.

If you’d like additional practice with the Pythagorean Theorem, take a moment to complete the quick practice below before closing out this lesson! ⚡

Practice: Pythagorean Theorem

Question 1 of 6: Calculate the length of the missing side:

$4$

$?$

$5$

Step 1. Identify the given and missing sides.

The small square inside the triangle points toward the hypotenuse (the longest side). The legs are the sides that form the right angle and L shape of the triangle.

Given Values:Missing:

Step 2. Imagine a square on each of the three sides of the triangle.

Step 3. Use Pythagorean's Theorem to find the missing side.

Pythagorean's Theorem tells us the areas of the smaller two squarescombined are equal to the area of the largest square. We can use this to find the missing side.

$a_{2}$$+$$b_{2}$$=$$c_{2}$

$_{2}$

$+$$b_{2}$$=$

$_{2}$

Great work! The unknown side is $3$.

When you feel like you've mastered this lesson, click for a celebration ⬇️!

Incredible job, look at you go! Thanks for checking out this lesson ☺️🙏. Where to next?