# Pythagorean Theorem Worksheet, Formula, and Examples

Get the free Pythagorean Theorem Worksheet and other resources for teaching & understanding solving the Pythagorean Theorem

### Key Points about the Pythagorean Theorem

• The Pythagorean Theorem describes the relationship between the sides of a right triangle.
• The formula for the Pythagorean Theorem is a^2 + b^2 = c^2, where c is the hypotenuse and a and b are the other two sides.
• The Pythagorean Theorem is essential for many fields, including architecture, engineering, and physics.

## What is the Pythagorean Theorem?

The Pythagorean Theorem is a formula for finding a missing side length of a right triangle. In order to answer how to do the Pythagorean Theorem you must understand the different sides of a right triangle. The side that is located across from the right angle is called the hypotenuse. The sides that are adjacent to the right angle are called the legs. The formula for the Pythagorean Theorem says that the hypotenuse squared is equal to the sum of the two legs squared. You can use the Pythagorean Theorem to find both a missing hypotenuse or a missing leg length.

The Pythagorean Theorem is a fundamental concept in mathematics that describes the relationship between the sides of a right triangle. It states that the square of the hypotenuse, the longest side of a right triangle, is equal to the sum of the squares of the other two sides. The theorem is named after the ancient Greek mathematician Pythagoras, who is credited with its discovery.

The Pythagorean Theorem formula is a^2 + b^2 = c^2, where c is the length of the hypotenuse and a and b are the lengths of the other two sides. This formula can be used to find the length of any side of a right triangle, as long as the lengths of the other two sides are known. Understanding the Pythagorean Theorem is essential for many fields, including architecture, engineering, and physics, where right triangles are commonly used.

Common Core Standard: 8.G.7
Related Topics: Parallel Lines Cut by a Transversal, Triangle Angle Sum, Exterior Angle of a Triangle, Volume of a Cylinder, Volume of a Cone, Volume of a Sphere

## The Pythagorean Theorem Formula

The Pythagorean Theorem is a fundamental concept in geometry that relates to the sides of a right triangle. It states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other two sides. This theorem is named after the ancient Greek mathematician Pythagoras, who first discovered it.

### Concept of Right Triangle

A right triangle is a triangle that has one angle that measures 90 degrees, also known as a right angle. The side opposite the right angle is called the hypotenuse. The other two sides are called the legs or the adjacent and opposite sides. The Pythagorean Theorem only applies to right triangles.

### Concept of Hypotenuse

The hypotenuse is the longest side of a right triangle and is always opposite the right angle. It is also the side that the Pythagorean Theorem formula solves for. The hypotenuse is denoted by the letter c.

### Concept of Sides

The other two sides of a right triangle are called the adjacent and opposite sides. The adjacent side is the side that is adjacent to the angle that is not the right angle. The opposite side is the side that is opposite to the angle that is not the right angle. These sides are denoted by the letters a and b, respectively.

The formula for the Pythagorean Theorem is:

c^2 = a^2 + b^2

where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

The Pythagorean Theorem formula can be used to find the length of any side of a right triangle, as long as the lengths of the other two sides are known. It can also be used to determine if a triangle is a right triangle, by checking if the sum of the squares of the lengths of the two shorter sides is equal to the square of the length of the longest side.

In summary, the Pythagorean Theorem is a fundamental concept in geometry that relates to the sides of a right triangle. It is named after the ancient Greek mathematician Pythagoras, who first discovered it. The formula for the Pythagorean Theorem is c^2 = a^2 + b^2, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides. The theorem can be used to find the length of any side of a right triangle and to determine if a triangle is a right triangle.

## How to do Pythagorean Theorem

To use the Pythagorean Theorem, one must first identify the right triangle. A right triangle is a triangle with one angle measuring 90 degrees. The side opposite the 90-degree angle is called the hypotenuse, and the other two sides are called the legs.

Once the right triangle is identified, one can use the Pythagorean Theorem to find the length of any of the sides. The theorem states that the square of the hypotenuse is equal to the sum of the squares of the legs. In other words, if a and b are the lengths of the legs, and c is the length of the hypotenuse, then c^2 = a^2 + b^2.

To find the length of one of the sides, one must rearrange the equation to solve for that side. For example, to find the length of the hypotenuse, one would take the square root of both sides of the equation, giving c = √(a^2 + b^2).

It is important to note that the Pythagorean Theorem only applies to right triangles. If the triangle is not a right triangle, then the theorem cannot be used.

To illustrate how to use the Pythagorean Theorem, consider the following example. Suppose a right triangle has legs of length 3 and 4. To find the length of the hypotenuse, one would use the equation c^2 = a^2 + b^2, where a = 3 and b = 4. Solving for c, one gets c = √(3^2 + 4^2) = √25 = 5. Therefore, the length of the hypotenuse is 5.

In summary, to use the Pythagorean Theorem, one must identify the right triangle and its legs. Then, one can use the equation c^2 = a^2 + b^2 to find the length of any of the sides. Finally, one must rearrange the equation to solve for the desired side.

## Pythagorean Theorem Definition

The Pythagorean theorem is a fundamental relation in Euclidean geometry that describes the relationship between the three sides of a right triangle. It states that the sum of the squares of the lengths of the two legs of a right triangle is equal to the square of the length of the hypotenuse. In other words, if a and b are the lengths of the legs and c is the length of the hypotenuse, then a² + b² = c².

This theorem is named after the ancient Greek mathematician Pythagoras, who is credited with its discovery and proof. However, it is likely that the theorem was known to other civilizations before him, such as the Babylonians and the Indians.

The Pythagorean theorem is often used in various fields of mathematics, science, and engineering. For example, it can be used to calculate the distance between two points in a two-dimensional plane or the length of a diagonal in a three-dimensional space. It is also used in trigonometry to find the values of sine, cosine, and tangent of an angle in a right triangle.

Overall, the Pythagorean theorem is a simple yet powerful mathematical concept that has numerous practical applications in various fields. ## 3 Simple Pythagorean Theorem Examples

The Pythagorean Theorem is a fundamental concept in geometry that is used to calculate the length of the sides of a right triangle.

1. The Pythagorean Theorem is the leg square plus the leg squared equals the hypotenuse square.
2. The legs are the sides that are touching the right angle.
3. The hypotenuse is the side that is across from the right angle.
4. You can use the two side lengths that you know to substitute into the Pythagorean Theorem.
5. Take the square root of the remaining side length to solve for the missing side.

Here are some examples of how the theorem can be applied in real-world situations.

### Example 1: Finding the Length of the Hypotenuse

Consider a right triangle with legs of length 3 cm and 4 cm. To find the length of the hypotenuse, we can use the Pythagorean Theorem, which states that the sum of the squares of the legs is equal to the square of the hypotenuse. In this case, we have:

• Leg 1: 3 cm
• Leg 2: 4 cm
• Hypotenuse: ?

Using the Pythagorean Theorem, we get:

3² + 4² = Hypotenuse²

9 + 16 = Hypotenuse²

25 = Hypotenuse²

Hypotenuse = √25 = 5 cm

Therefore, the length of the hypotenuse is 5 cm.

### Example 2: Checking if a Triangle is a Right Triangle

Suppose we have a triangle with sides of length 6 cm, 8 cm, and 10 cm. To check if this triangle is a right triangle, we can use the Pythagorean Theorem. If the sum of the squares of the two shorter sides is equal to the square of the longest side, then the triangle is a right triangle.

In this case, we have:

• Side 1: 6 cm
• Side 2: 8 cm
• Hypotenuse: 10 cm

Using the Pythagorean Theorem, we get:

6² + 8² = 10²

36 + 64 = 100

100 = 100

Since the sum of the squares of the two shorter sides is equal to the square of the longest side, this triangle is a right triangle.

### Example 3: Finding the Distance Between Two Points

Suppose we have two points, A and B, in a two-dimensional coordinate system. The coordinates of point A are (3, 4), and the coordinates of point B are (6, 8). To find the distance between these two points, we can use the Pythagorean Theorem.

We can treat the line connecting points A and B as the hypotenuse of a right triangle. The length of the horizontal leg is the difference between the x-coordinates of the two points, and the length of the vertical leg is the difference between the y-coordinates of the two points.

In this case, we have:

• x-coordinate difference: 6 – 3 = 3
• y-coordinate difference: 8 – 4 = 4

Using the Pythagorean Theorem, we get:

3² + 4² = Distance²

9 + 16 = Distance²

25 = Distance²

Distance = √25 = 5

Therefore, the distance between points A and B is 5 units.

## 5 Quick Pythagorean Theorem Practice Problems

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Pythagorean Theorem Quiz

Click Start to begin the practice quiz!

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Find the missing sign length using the Pythagorean Theorem. 2 / 5

Find the missing sign length using the Pythagorean Theorem. 3 / 5

Find the missing sign length using the Pythagorean Theorem. 4 / 5

Find the missing sign length using the Pythagorean Theorem. 5 / 5

Find the missing sign length using the Pythagorean Theorem. 0%

## Pythagorean Theorem Proof

The Pythagorean Theorem is a fundamental concept in mathematics that describes the relationship between the sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem is named after the ancient Greek mathematician Pythagoras, who is credited with its discovery.

There are many different proofs of the Pythagorean Theorem, each utilizing a different approach and level of complexity. Some proofs are geometric in nature, while others are algebraic. Some rely on Euclidean geometry, while others use trigonometry. However, all of the proofs arrive at the same conclusion: the Pythagorean Theorem is true and can be used to solve a wide range of mathematical problems.

One of the most famous proofs of the Pythagorean Theorem is the one attributed to Pythagoras himself. This proof relies on the fact that the area of a square is equal to the length of its sides squared. By constructing squares on each of the three sides of a right triangle and comparing their areas, Pythagoras was able to derive the formula for the Pythagorean Theorem.

Another popular proof of the Pythagorean Theorem is the algebraic proof. This proof involves using algebraic equations to manipulate the sides of a right triangle until the Pythagorean formula is derived. This proof is often used in high school mathematics courses and is a good introduction to the concept of mathematical proof.

There are also many other proofs of the Pythagorean Theorem, including geometric proofs, algebraic proofs, and trigonometric proofs. Each of these proofs offers a unique perspective on the theorem and can be used to deepen one’s understanding of the concept.

Overall, the Pythagorean Theorem is a fundamental concept in mathematics that is used in a wide range of applications. Its many proofs offer a glimpse into the beauty and complexity of mathematics and can be used to inspire further study and exploration.

## Converse of the Pythagorean Theorem

The converse of the Pythagorean Theorem is a statement that is derived from the original theorem. It states that if the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. In other words, if c² = a² + b², then the triangle is a right triangle.

This converse theorem is useful in determining whether a triangle is a right triangle or not. It is often used in geometry problems to find missing side lengths or angles of a triangle. If the converse of the Pythagorean Theorem is true, then the triangle is a right triangle. If it is not true, then the triangle is not a right triangle.

To better understand the converse of the Pythagorean Theorem, let’s consider an example. Suppose we have a triangle with sides of length 3, 4, and 5. We can use the Pythagorean Theorem to confirm that this triangle is a right triangle: 3² + 4² = 5². Therefore, the triangle is a right triangle.

Conversely, if we have a triangle with sides of length 5, 12, and 13, we can use the converse of the Pythagorean Theorem to confirm that this triangle is also a right triangle: 5² + 12² = 13². Therefore, the triangle is a right triangle.

It is important to note that the converse of the Pythagorean Theorem only works in one direction. Just because a triangle is a right triangle does not mean that it satisfies the conditions of the converse theorem. Therefore, it is essential to use both the Pythagorean Theorem and its converse together to determine whether a triangle is a right triangle or not.

## Pythagorean Theorem Word Problems

Pythagorean Theorem is a useful mathematical theorem that helps in solving problems related to right triangles. It states that the squares of the lengths of the legs of a right-angled triangle add up to the square of the length of the hypotenuse. The Pythagorean equation a² + b² = c² can be used to calculate the value of any of the three sides of a right triangle, given the other two sides.

Pythagorean Theorem is often used to calculate the distance between two points in geometry. For example, if two points are given in a coordinate plane, the distance between them can be calculated using the Pythagorean equation. Similarly, if the lengths of two sides of a right triangle are given, the length of the third side can be calculated using Pythagorean Theorem.

Pythagorean Theorem can also be used to solve problems related to similar triangles. If two triangles are similar, the ratio of their corresponding sides is the same. Therefore, if the length of one side of a similar triangle is known, the lengths of the other sides can be calculated using the ratio of the corresponding sides.

Pythagorean Theorem word problems are a great way to practice solving problems using this theorem. These problems usually involve finding the length of a missing side of a right triangle or the distance between two points. Here are a few examples of Pythagorean Theorem word problems:

• A ladder is leaning against a wall. The base of the ladder is 4 feet away from the wall and the ladder makes a 30-degree angle with the wall. How long is the ladder?
• A right triangle has legs of length 3 and 4. What is the length of the hypotenuse?
• A rectangular field measures 30 meters by 40 meters. What is the length of the diagonal of the field?

In order to solve Pythagorean Theorem word problems, it is important to identify the right triangle and label its sides. Once the sides are labeled, the Pythagorean equation can be used to calculate the length of the missing side. A calculator can be used to simplify the calculation and obtain the final answer.

In conclusion, Pythagorean Theorem is a mathematical theorem that is useful in solving problems related to right triangles and similar triangles. Pythagorean Theorem word problems are a great way to practice using this theorem and can involve finding the length of a missing side of a right triangle, the distance between two points, or the length of the diagonal of a rectangle. By identifying the right triangle and labeling its sides, the Pythagorean equation can be used to solve these problems.

## How to Solve Pythagorean Theorem FAQ

### What is the formula for finding the hypotenuse of a right triangle?

The formula for finding the hypotenuse of a right triangle is derived from the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. Therefore, the formula for finding the hypotenuse is:

c = √(a² + b²)

where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

### How do you use the Pythagorean theorem to find the length of one side of a right triangle?

To find the length of one side of a right triangle using the Pythagorean theorem, you need to know the lengths of the other two sides. Once you have these lengths, you can use the formula:

a = √(c² – b²)

or

b = √(c² – a²)

where a and b are the lengths of the two sides, and c is the length of the hypotenuse.

### What is the Pythagorean theorem used for in real life?

The Pythagorean theorem is used in a variety of real-life applications, including construction, architecture, engineering, and physics. For example, it can be used to calculate the distance between two points in space or to determine the length of a ladder needed to reach a certain height.

### What is the Pythagorean theorem and how is it used?

The Pythagorean theorem is a fundamental theorem in geometry that describes the relationship between the sides of a right triangle. It is used to calculate the length of the hypotenuse or one of the other two sides of a right triangle.

### How do you prove the Pythagorean theorem?

There are many ways to prove the Pythagorean theorem, including algebraic proofs, geometric proofs, and visual proofs. One of the most famous proofs is the one using squares, where a square is constructed on each of the three sides of a right triangle, and the area of the largest square is shown to be equal to the sum of the areas of the other two squares.

### What are some examples of real-world applications of the Pythagorean theorem?

Some examples of real-world applications of the Pythagorean theorem include calculating the distance between two points on a map, determining the length of a diagonal on a TV or computer screen, and calculating the height of a building using a laser rangefinder.

### What is the Pythagorean theorem 3 and 4?

The Pythagorean theorem 3 and 4 refers to the fact that the sides of a right triangle can be any multiple of the 3-4-5 triangle. In other words, if one side of a right triangle is 3 units and another side is 4 units, then the length of the hypotenuse is 5 units.

### Can you use Pythagorean Theorem on a non right triangle?

No, the Pythagorean theorem can only be used on right triangles. For non-right triangles, other formulas and methods must be used to calculate the length of the sides.

### What is the Pythagorean formula in simple terms?

The Pythagorean formula is a mathematical equation used to calculate the length of the hypotenuse or one of the other two sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

## Watch the video where we complete our Pythagorean Theorem Worksheet

Watch our free video on how to do Pythagorean Theorem. This video shows how to solve problems that are on our free Pythagorean Theorem worksheet that you can get by submitting your email above.

Watch the free Pythagorean Theorem video on YouTube here: Pythagorean Theorem Worksheet

Video Transcript:
This video is about the Pythagorean theorem and we are using the pythagorean theorem worksheet to show you a few pythagorean theorem examples.

The directions on our Pythagorean theorem practice worksheet say to find the missing side length. We have our triangle here at number one and we have two sides that are given to us, we have this side that’s given to us on this side that’s given to us now. In the case of these two sides they are both adjacent to the right angle, now all sides that are adjacent to the right angle are known as legs. That’s a leg and then this over here is a leg the side across from the right angle is known as the hypotenuse. When we do our Pythagorean theorem which is a squared plus B squared equals C squared.

In the case of this example you could say or really in the case of all examples a is always going to be a leg. We’ll have a leg squared and then B is always going to be a leg  this would be the other leg squared is equal to the hypotenuse squared. This will always be either A or B and this will always be either A or B and this will always be C no matter what. In the case of our example here we’re gonna say that a is 10. We’re gonna substitute in a are 10 for a. 10 goes in there and we’re going to say that B is 24. We’re going to substitute in 24 for B and then C we don’t know  we’re going to just leave C as it is C squared. Now in order to simplify what we’re going to do is we’re going to do 10 squared and then we’re going to do 24 squared. 10 times 10 or 10 squared is 100 and then 24 times 24 is 576 and then over here we have C squared and we just that is C squared.

Then the next step is to add these two together which obviously will be 676 equals C squared and then the final step is to get rid of this square here. We have C squared we have to get rid of that and we’re going to take the square root because the opposite of squaring something is taking the square root. We’re going to do the square root on the side whatever you do to one side you have to do the other so we’re also going to do the square root on this side on this side. The square and the square root cancel and you’re left with just C and then on this side the square root of 676 is 26. The answer to our missing side length is 26.

We’re going to move on to our next problem our next problem is a little bit different because we are missing a leg instead of the hypotenuse. Here is an example from our pythagorean theorem missing side worksheet. If you look the first thing you might want to do is label the sides of this triangle. In the case of this triangle we have a leg here because it’s adjacent to the right angle and then we have a leg here because it’s adjacent to the right angle and then this side over here has to be the hypotenuse because it is across from the right angle.

This is how our triangle is in terms of labeling and we will have to solve the pythagorean theorem missing leg. This for the Pythagorean theorem we know that a leg has to be a and B and the hypotenuse has to be C. When we set up our Pythagorean theorem this time we know that it’s a squared plus B squared equals C squared but this time we know a is 35. It’s 35 squared and we don’t know B this time so we’ll just leave it as B squared but we do know C which is 37. It’ll be 37 squared the next step is to square 35. 35 times 35 is 1225 plus B squared we just leave as B squared because we don’t know and then 37 squared is 1,000 369.

Then the next step to solve this is to subtract 1225 from 1369 these will cancel you bring down your b squared on this side and then 1369 minus 1225 is 144. Then the last step that you have to solve is you have to get rid of this square the opposite of squaring something is taking the square root. We’re going to take the square root of both sides you end up with the square root of 144 which is 12. The answer to our missing side length in this problem is 12. You can try the practice practice problems by downloading the free pythagorean theorem worksheets above.

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