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Pythagorean Theorem Worksheets

These printable worksheets have exercises on finding the leg and hypotenuse of a right triangle using the Pythagorean theorem. Pythagorean triple charts with exercises are provided here. Word problems on real time application are available. Moreover, descriptive charts on the application of the theorem in different shapes are included. These handouts are ideal for 7th grade, 8th grade, and high school students. Kick into gear with our free Pythagorean theorem worksheets!

Identifying Right Triangles

Apply Pythagorean theorem to identify whether the given triangle is a right triangle. Each printable worksheet consists of six problems.

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Pythagorean Theorem Chart

These descriptive charts explain the Pythagorean theorem with an illustration. These pdfs emphasize the relation of the theorem derived as an equation.

Pythagorean Triple

A set of three numbers is given in each problem. Grade 7 and grade 8 students need to apply the theorem and identify whether the set of numbers forms a Pythagorean triple.

Pythagorean Triple Chart

This section comprises of Pythagorean triple sets up to 100. Besides, Pythagorean triple formulas with examples are provided in the charts.

Unknown Side of a Right Triangle

Apply Pythagorean theorem to find the unknown side of the right triangle. Round the answer to the nearest tenth.

Unknown Length: Shapes

Apply the Pythagorean theorem to find the unknown length of each shape in these printable worksheets. Round the answer to the nearest tenth.

Word Problems | Level 1

Brighten your math class with this bundle of real-life word problems based on the Pythagorean Theorem. Solve each word problem by finding the missing hypotenuse of the right triangle and rounding off the answer to the nearest tenth.

Word Problems | Level 2

Presenting word problems with clear illustrations, these pdf worksheets require high school students to plug in the known values into the equation form of the Pythagorean Theorem and figure out the unknown side of the right triangle.

Related Worksheets

» Pythagorean Inequality Theorem

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» Triangle Inequality Theorem

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Pythagorean Theorem Worksheet: 4 Levels

Looking for the perfect Pythagorean theorem worksheet? Need some right-angle triangle practice? Look no further!

The theorem may be short and seem simple, but there are many types of questions you could face in an exam.

You will use this theorem if you continue studying math at a higher level. It’s a useful equation to keep in your math tool kit !

Applying the theorem with confidence is helpful for many people in the world.

The Pythagorean theorem really does have lots of real-life uses. You may use it for crafts, in construction, for baking, gardening, or choosing a smartphone. Without it, GPS location wouldn’t be a thing!

Each worksheet has a different focus. You might have one particular area you’d like to improve on, or perhaps you’re looking for a complete crash course.

Either way, don’t forget to show your working out, and start at the right skill level – each Pythagorean theorem worksheet is given in order of difficulty!

Worksheets The Man, the Legend: Pythagoras of Samos Who ACTUALLY Wrote the Pythagorean Theorem? What is the Theorem Other Theories and Discoveries Attributed to Pythagoras How Many Solutions? Solutions When a = 2 Special Solutions: Triples Properties of Triples A Brief Example Families of Triples A Briefer Example To Sum Up (Pun Intended!)

Free Printable Pythagorean Theorem Worksheet With Answers

Level 1: identify the right-angle triangles.

Level 2: Identify the Pythagorean Triples

Level 3: Find the Missing Side Length of Right-Angle Triangles

Level 4: Real-World Pythagorean Theorem Word Problems (2 Pages)

The Man, the Legend: Pythagoras of Samos

The Pythagorean theorem is named after the Greek mathematician and philosopher, Pythagoras, who was born around 570 BC.

His home was the small Greek island Samos which is in the Mediterranean Sea, near Turkey.

Although he grew up in a beautiful paradise, there was political conflict. He traveled to Egypt and Persia as a young man before settling in southern Italy.

In Crotone, Italy, he established a school of philosophy and science. His students were called his disciples, his school like a family – a brotherhood.

In fact, the group he established was called The Brotherhood of Pythagoreans.

The Brotherhood had a structured life of exercise and study. They believed that to live forever in their second life , they needed to be physically perfect.

Pythagoras and his disciples, as they were called, were incredibly disciplined. They wanted to keep their bodies “morally pure”.

He was, by most accounts, a wacky guy – he always put his right shoe on first, never ate beans, didn’t travel by public roads, and encouraged his students to be silent wherever possible!

None of his writings have survived. That isn’t too surprising, given that he lived almost 2600 years ago!

Pythagoras is a man of mystery. Without his records, everything we know about him comes from what his disciples and other scholars wrote about him.

Imagine trying to find out what someone else is like, and what discoveries they have made, from their friends and family’s posts on social media. You wouldn’t get all the detail you need!

It’s known that he taught a special “Pythagorean Way of Life,” based on religion and ritual. Interestingly though, his beliefs were shaped by mathematics.

After he died in roughly 490BC, many people glorified Pythagoras and considered him God-like.

Apparently, he performed miracles: he killed a dangerous snake by biting it, he was seen in two cities at the same time, a river spoke to him and his thigh was golden.

Click here to go back to the Pythagorean theorem worksheet .

Who ACTUALLY Wrote the Pythagorean Theorem?

The Pythagorean Theorem is what Pythagoras is most famous for – it’s probably the first thing you think of when you hear his name.

But did Pythagoras himself really think of the theorem? Historians don’t know! It’s possible that one of his disciples wrote it after Pythagoras died, and named it after him.

The Pythagoreans were secretive. They came up with many proofs of important mathematical ideas but didn’t share their discoveries with anyone else.

If Pythagoras really did write the famous theorem, he didn’t tell the world about it!

What is the Pythagorean Theorem?

First, a right-angled triangle is a triangle that has an internal angle of 90°. It looks a bit like a ladder leaning between the floor and a wall.

Second, the hypotenuse of a right-angled triangle is its longest side. It is always the diagonal and never touches the right-angle. It would be the ladder in this picture!

“In a right-angled triangle, the square of the hypotenuse equals the sum of squares of the other two sides”.

Don’t let the word ‘theorem’ put you off! It boils down to one equation:

Where a and b are the smaller side lengths and c is the length of hypotenuse .

If you have two lengths of a right-angled triangle, you can find the third using the formula.

Other Theories and Discoveries Attributed to Pythagoras

He was a well-rounded guy!

In Ancient Greek society, Pythagoras was credited with being the first to teach that the Earth is a sphere.

He was also one of the first to divide the world into five climatic zones – two cold zones at the poles, a tropical zone across the equator, and milder zones between them.

Pythagoras believed that math and abstract thought are the basis of morality and science. His teachings in philosophy inspired famous philosophers Plato and Aristotle.

Pythagoras started the idea of a number system, founded on his belief that numbers make up everything in the world.

He believed that all numbers are rational – that they can be written as fractions – and that they were beautiful in their simplicity.

The Pythagoreans were shocked when they discovered \(\sqrt{2\ }\), an irrational number that cannot be written as a fraction.

Pythagoras linked math to music when he heard blacksmiths’ hammers hitting metal. He realized that the pitch of the sound made was related to the size of the hammer.

Certain pitches sound nice when played together – they are in consonance. Other pitches clash, making a nasty sound – they are in dissonance.

Playing notes with special intervals between their pitches produces harmonies. Thank Pythagoras next time you hear singers harmonizing!

How Many Solutions?

There are infinitely many solutions to the equation a 2 +b 2 =c 2

This becomes clear if one of the values is fixed. Let’s suppose that a=2 and see how many pairs of b and c values make sense.

This equation gives the relationship between b and c . Any value of c>2 has a corresponding value of b which satisfies the Pythagorean theorem because the radical is positive.

If c<2 , the equation has imaginary, or complex numbers .

For example, if we let c=1 , then

which isn’t a real number, because no real number can multiply by itself to make a negative number!

Solutions When a=2

We have just said that you can pick any value of c>2 and plug this into the equation \(b=\sqrt{c^2 – 4}\) to complete the solution.

Let’s give this a try:

If you stop to think about it, what would a right-angle triangle with two sides of the same length look like? Let’s use math to find out.

We’re going to let a=2 and c=2 , and put them into the equation to see what happens.

Hmmm, a right-angle triangle with one side of zero length? The math has spoken – there is no such thing!

There are infinitely many values of c to pick from, so there are infinitely many corresponding values of b , so there are infinitely many solutions to the Pythagorean Theorem when a=2 !

This thinking can be applied to any fixed value of a . There are infinitely many solutions for each fixed value of a , and there are infinitely many a ’s to pick from.

In short, that’s a lot of solutions!

Special Solutions: Pythagorean Triples

The possibilities are literally endless when it comes to solving this equation.

It’s also interesting to study a subset of solutions, called Pythagorean triples .

These are triplets of positive integers a , b and c which solve the equation a 2 +b 2 =c 2 .

The first – and most famous! – example is 3 2 +4 2 =5 2 which, when used for measuring, is often referred to by tradespeople as 3, 4, 5.

Pythagorean triples are easier to work with because no radicals are involved. However, there is less freedom when creating them.

These triplets aren’t constructed by picking two variables and solving for the third – because of the special integer property, we must be more sophisticated!

Properties of Pythagorean Triples

The integer triplets \((a,b,c)\) fit into one of two categories: either all three numbers are even, or one is even and the other two are odd.

This is because the squares of even numbers are even, and the squares of odd numbers are odd.

Adding to this property, some equations help us construct a valid Pythagorean triple.

Begin by picking two positive integers. Call the bigger number x , and the smaller number y . It’s important that x and y are not equal to each other.

Substituting x and y into the following equations will give your triple:

Set x=10, and y=5. This is valid because \(x\neq y\) and \(x>y\).

The Pythagorean triple is

It’s written as:

In this case, two of the numbers are odd and one is even so it’s in the second category.

We can check that the triple satisfies the Pythagorean theorem by simply substituting in a , b , and c to find that:

Families of Triples

Equations can be manipulated by doing the same thing to both sides. Even though it has a fancy name, Pythagoras’ equation is no different!

If you multiply both sides of an equation by a positive integer, it’s still valid.

This idea is super useful – multiplying a triple by a positive integer will always give another triple!

Multiplying a triple is the same as multiplying both sides of the corresponding Pythagorean equation.

Each number in the triple is multiplied by a positive integer, giving another positive integer.

Think of it as scaling up the triangle, dragging it by the corner as you would with a photo or textbox on a computer.

Dividing triples is a dangerous game – if the divisor is not a factor of each number in the triple, the resulting numbers will not be integers so a Pythagorean triple will not be made. Disaster!

Every triple can be scaled upwards but not all can be scaled downwards.

When multiplying triples, you’ve got a lot of options. Infinitely many, in fact!

This means that there is an infinite family of triples, related to one another by multiplication.

In each ‘family’ of related triples, the smallest is given a special name.

A Pythagorean triple is called primitive if the greatest common divisor of a , b and c is 1.

Primitive triples are the simplest elements in their families – they cannot be divided to produce a smaller triple.

Let’s return to the previous triple we found: (75, 100, 125) .

This triple is not primitive because of the greatest common divisor, or GCD , which is 25, not 1.

Dividing each number by 25 gives another Pythagorean triple:

The division within the exponent looks a little suspicious, but we aren’t breaking any rules.

It is the same as dividing both sides by 25 2 , which is allowed!

To Sum Up (Pun Intended!)

I hope you’ve enjoyed learning some history along with your math. Pythagoras is a legendary figure, shrouded in mystery, but his theorem doesn’t have to be.

After using a Pythagorean theorem worksheet, you should have the confidence to identify right-angled triangles, find missing side lengths, identify Pythagorean triples, solve real-life problems, and use the theorem in trickier shapes.

Each worksheet uses the same amazing little equation a 2 +b 2 =c 2 , discovered around 2500 years ago.

Pythagoras is most famous for this theorem, but he had a busy life! One of his achievements was creating a circle of scholars called The Brotherhood of Pythagoreans.

They were influential in Ancient Greek society – Pythagoras’s work in philosophy inspired Plato and Aristotle, and the Brotherhood made many discoveries.

To count the number of solutions to his equation would take a long time – an infinite time, in fact! There is also an infinite number of integer solutions, called Pythagorean triples.

These triples are related to each other by multiplication . The first triple in each related ‘family’ is called primitive.

Once you’ve mastered using the theorem, you might enjoy learning about other mathematicians! The story of Fermat’s last is particularly interesting and begins with a similar equation…

If you have a question or comment to leave below, don’t be shy! It’s helpful to learn from each other.

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Geometry Worksheets to Practice Using the Pythagorean Theorem

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The Pythagorean Theorem is believed to have been was discovered on a Babylonian tablet circa 1900-1600 B.C.

The  Pythagorean Theorem  relates to the three sides of a right triangle . It states that c2=a2+b2, C is the side that is opposite the right angle which is referred to as the hypotenuse. A and b are the sides that are adjacent to the right angle.

The theorem simply stated is: the  sum of the areas  of two small squares equals the area of the large one.

You will find that the Pythagorean Theorem is used on any formula that will square a number. It's used to determine the shortest path when crossing through a park or recreation center or field. The theorem can be used by painters or construction workers, think about the angle of the ladder against a tall building for instance. There are many word problems in the classic math textbooks that require the use of the Pythagorean Theorem.

History Behind Pythagorean's Theorem

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Hippasus of Metapontum was born in the 5th Century BC. It is believed that he proved the existence of irrational numbers at a time when the Pythagorean belief was that whole numbers and their ratios could describe anything that was geometric. Not only that, they didn't believe there was a need for any other numbers .

The Pythagoreans were a strict society and all discoveries that happened had to be directly credited to them, not the individual responsible for the discovery. The Pythagoreans were very secretive and did not want their discoveries to 'get out' so to speak. They considered whole numbers to be their rulers and that all quantities could be explained by whole numbers and their ratios. An event would happen that would change the very core of their beliefs. Along came Pythagorean Hippasus who discovered that the diagonal of a square whose side was one unit could not be expressed as a whole number or a ratio.

What Is the Hypotenuse?

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Simply put, the hypotenuse of a right triangle is the side opposite the right angle. It is sometimes referred to by students as the long side of the triangle. The other two sides are referred to as the legs of the triangle. The theorem states that the square of the hypotenuse is the sum of the squares of the legs. 

The hypotenuse is the side of the triangle where C is. Always understand that the Pythagorean Theorem relates the areas of squares on the sides of the right triangle

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Free Printable Pythagorean Theorem worksheets

Pythagorean Theorem worksheets: Discover an extensive collection of free printable math resources, designed to help students and teachers explore the fascinating world of the Pythagorean Theorem.

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Pythagorean Theorem worksheets are an essential resource for teachers looking to enhance their students' understanding of Math and Geometry concepts, specifically Triangle Theorems. These worksheets provide a variety of problems that challenge students to apply their knowledge of the Pythagorean Theorem in different contexts, helping them to develop a deeper comprehension of the relationships between the sides of a right-angled triangle. By incorporating these worksheets into their lesson plans, teachers can effectively engage students in hands-on learning activities that reinforce key mathematical concepts. Furthermore, these worksheets can be easily adapted to suit the needs of students at different grade levels, making them a versatile and valuable addition to any Math or Geometry curriculum.

Quizizz is an excellent platform for teachers to access a wide range of Pythagorean Theorem worksheets, as well as other Math and Geometry resources. This interactive platform offers a variety of engaging quizzes and activities that can be used to supplement traditional worksheets, providing students with a more dynamic and enjoyable learning experience. Teachers can easily customize the content on Quizizz to align with their specific grade level and curriculum requirements, ensuring that the material is both relevant and challenging for their students. Additionally, Quizizz offers valuable insights into student performance, allowing teachers to identify areas where students may need additional support or practice. By incorporating Quizizz into their teaching strategies, educators can create a more engaging and effective learning environment for their students, ultimately leading to greater success in mastering the Pythagorean Theorem and other essential Math and Geometry concepts.

The Converse of the Pythagorean Theorem

New york state common core math grade 8, module 7, lesson 16.

Lesson 16 Student Outcomes

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Pythagorean Theorem Worksheets

When a triangle has a right side (measuring ninety degrees) and you extend each side with a square, the biggest square has the same exact are as the sum of the squares of the other two sides. If we labelled the longest side c and the other two sides (legs) a and b, we could sum up that phenomena with the equation a 2 + b 2 = c 2 . We term the longest side, that opposite the right angle, the hypotenuse. As you gather by the algebra, if you know the measures of any two sides of a right triangle, you can use this theorem to determine the third side. Pythagorean Theorem lends itself to all form of building and construction because it allows you to easily determine the length of any material that we using as long as you have two materials that meet at a right angle. Sailors use this all the time to find the shortest route between two points. You will find many applications in the field of surveying when cartographers and engineers are constructing maps that indicate steepness and slopes of areas.

Aligned Standard: Grade 8 Geometry - 8.G.B.8

• Applying the Pythagorean Theorem Step-by-Step Lesson - The side that is missing is defined as the hypotenuse. We can tell this because it is directly opposite the right angle.
• In Action Guided Lesson - Example: Given a right triangle with a hypotenuse of 27 and a leg of 17, what is the measure of the missing leg?
• Guided Lesson Explanation - If this is a right triangle it will work out when the Pythagorean Theorem is applied to it. This means it will flow in the formula: leg 2 + leg 2 = hypotenuse 2
• Are They Right Triangles? - Determine the classification of these shapes.
• Skills Review - We explore every way this topic can be looked at.
• Answer Keys - These are for all the unlocked materials above.

Review Sheets

The coordinate graph makes it much more understandable than just labeling sides.

• Review 1 - We will look this from the diagram perspective and in word form.
• Review 2 - Another go at this series of skills.

Homework Sheets

• Homework 1 - We will see if these shapes will work for us.
• Homework 2 - Another approach at the first version.
• Homework 3 - All of these shapes are much more narrow.
• Homework 4 - You will find decimal lengths to work with.

Practice Worksheets

We start to find random distances between points.

• Practice 1 - You will work with some really scaled down sides.
• Practice 2 - We orientate these shapes from all directions.
• Practice 3 - This is a basic look at this skill.
• Practice 4 - More decimal lengths to work with.

Math Skill Quizzes

Here are more like problems to help you master this topic.

• Quiz 1 - Use the Pythagorean theorem to find the length of the missing side of the right triangle.
• Quiz 2 - Apply this formula to see if you can determine that missing side.
• Quiz 3 - Time break out the decimal form one more time.

How is the Pythagorean Theorem Used Every in Life?

The history behind how this theorem was conceived is almost fiction-like. Pythagoras was a Greek mathematician, among other things. He was born around 500 B.C. He founded a group called the Brotherhood of Pythagoreans which was devoted towards the study of mathematics. The theorem is credited to Pythagoras, but historians do strongly suggest that it was the effort of this group.

More than 2,500 years later this math still holds up and we use this theorem in our daily lives without even realizing it. It goes without saying that most forms of architecture and construction projects heavily rely on this to form straight lines in their buildings. It also is the foundation of how the angle of roofing is pitched on almost all structures. All the beams and materials used are dependent on this calculation. Something as simple as determining how to optimal position ladders against a wall for safety are calculated in this manner as well.

The Brotherhood of Pythagoreans probably never thought that large water vessels would depend on their math to circumnavigate the Earth's oceans. From very short distances to long, understanding the best path is often a product of this theorem. It also helps ships Captains determine how to interest or disrupt enemies in battle. Watch just about any war movie that involves a ship or a submarine and you will see mathematical compasses and rulers putting this to work.

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