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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Maths Worksheet for Kids | Pythagorean Theorem Practice Worksheet - Download Free PDF with Solutions
If you want to learn all about right-angled triangles and find the leg and base values best, you can download and refer to the Pythagorean Theorem Practice Worksheet from Vedantu. These worksheets are available in PDF format to guide the students about basic concepts related to the Pythagorean theorem. Students of KG-3 can download the worksheets and practice the questions from there to gain better knowledge about the topics in the first place.
The printable PDF worksheets have different exercises students can solve to build a strong conceptual foundation about the theorem. There are many different types of Pythagoras Theorem Questions in these exercises to make sure that students can get a good grasp of the topic in the first place. Hence, students should download the worksheets and learn the type of questions provided there.
Access Worksheet for Maths KG-2 Pythagorean Theorem Practice
In Maths, the Pythagorean theorem states that “In a right triangle, the square of the hypotenuse equals the sum of squares of the remaining two sides.”
Let’s solve some questions for further understanding -
Identify the length of the hypotenuse in the given triangle -
Image - An Image of a Triangle
Identify the length of the perpendicular in the given triangle -
Image - An Image of a Triangle
Is the given figure a right-angled triangle?
Identify the length of the base in the given triangle -
Image - an Image of a Triangle
Use Pythagoras' Theorem to check whether the given triangle is a right-angle triangle or not.
Find the value of x.
Pythagoras' theorem is applicable in which type of triangles?
[Acute triangles, right-angled triangles, or obtuse triangles]
Is the given triangle right-angled?
Find the length of the hypotenuse in the given triangle.
Find the length of the hypotenuse in the given triangle using Pythagoras' theorem.
Image - an image of a triangle
Does the Pythagorean theorem apply to right triangles?
Is the following Pythagorean triplet correct?
Solve for an in the given triangle -
Solve for c in the given triangle -
8.1m, the hypotenuse length in the given triangle is 8.1m.
4cm 6cm, the perpendicular length in the given triangle can be 4cm or 6cm.
Yes, the given triangle is right-angled.
9cm or 12cm. The length of the base in the given triangle can be 9cm or 12cm.
By pythagoras theorem, a 2 + b 2 = c 2
Let a=9 b=5 c=10.3
81 + 25 = 106 ⇔ 10.3 × 10.3 = 106 approx.
Therefore, LHS=RHS
Hence, the given triangle is a right-angle triangle.
Given a=6 b=8
36 + 64 = c 2 ⇒ 100 = c 2
Hence, the value of x is 10.
Pythagoras' theorem is applicable only in right-angled triangles.
No, the given triangle is not right-angled.
9.3cm the hypotenuse length in the given triangle is 9.3cm.
By Pythagoras' theorem, a 2 + b 2 = c 2
Let a=3 b=7
9 + 49 = c 2 ⇒ 58 = c 2
Hence, the hypotenuse length is √58.
Yes, the Pythagorean Theorem applies to all right-angled triangles.
By Pythagoras theorem, a 2 + b 2 = c 2
Let a=3 b=4 c=5
Hence, 3, 4 and 5 are correct Pythagorean triplets.
Given, b=12 and c= 15
a 2 + b 2 = c 2
a 2 + 144 =225
Subtract 144 from each side to get:
144 - 144 + a 2 = 225 - 144
a 2 = 225 - 144 ⇒ a 2 = 81
Given, a=3 b=4
By Pythagoras’ theorem,
32 + 42 = c 2
3×3 + 4×4 = c 2
9 + 16 = c 2
Did You Know?
Were you aware that Pythagoras, the person who is credited for discovering the Pythagorean Theorem, claimed to have already lived four lives? Yes, the renowned mathematician claimed that he remembered the previous four lives that he had lived. Apart from that, he was also supposedly a great friend to animals and loved them very much.
Importance of Pythagorean Theorem Practice Questions for Students
Students of KG-3 will have to complete their Maths syllabus to gain a good understanding of the topics. In that process, the worksheets for the Pythagorean theorem can help them out. The concepts have been explained very well in the worksheets. Additionally, there are some very important questions hand-picked by the experts at Vedantu for a better student learning experience. They can score good marks on these worksheets.
The worksheets contain numerical problems, Pythagorean theorem word problems, real-time problems, and much more. By repeatedly solving the questions from the worksheets, students will gain a good understanding of the different topics and processes explained here. Students can rely on the authenticity of the worksheets as the questions are on par with the school board's standards. Students will also find the Pythagoras theorem chart depictions and learn how to create them.
Benefits of Pythagoras Theorem Practice Questions
Students can understand a lot about the topic by solving the questions from the worksheets.
Finding out the values related to the aspects of a right-angled triangle has been explained so that students can learn how to solve these questions.
The experts at Vedantu have provided an easy and simple explanation for all the solutions. Students will be able to understand the solutions without any additional effort easily.
Overall, the worksheets from Vedantu are resourceful study materials that will help the students prepare for their exams.
Students can rely on the authenticity of the worksheets and the solutions as they are entirely accurate and contain a lot of information that will be useful for students.
Download Pythagorean Theorem Worksheets From Vedantu
Your chance to download the Pythagorean triples examples with answers is right here. Get your hands on these helpful worksheets and better understand the concepts. These worksheets will help you practice appropriately so you can score some good marks in the examinations.
FAQs on Maths Worksheet for Kids- Pythagorean Theorem Practice Worksheet
1. What is the Pythagorean theorem?
According to the Pythagoras theory, the square of the hypotenuse of a right-angled triangle will be the sum of the squares of the base and the perpendicular of the triangle.
2. Where can I use the Pythagoras theorem?
You can use the Pythagoras theorem to solve different questions related to the area and perimeter of a right-angled triangle.
3. Can I score good marks in Maths with the Pythagorean theorem worksheet?
Yes, you can definitely score some good marks in your exams if you practice solving the questions regularly from the worksheets for the Pythagorean theorem.
4. How can I use the practice problems to score good exam marks?
After downloading the Pythagorean theorem practice problems, students can go ahead and practice the problems again and again, to get better at them.
5. Are the worksheets utterly free of cost?
Yes, the Pythagorean theorem practice worksheets are free of cost and can be downloaded from the official website of Vedantu.
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The Pythagorean Theorem Date_____ Period____ Do the following lengths form a right triangle? 1) 6 8 9 No 2) 5 12 13 Yes 3) 6 8 10 Yes 4) 3 4 5 Yes ... Create your own worksheets like this one with Infinite Pre-Algebra. Free trial available at KutaSoftware.com. Title: Pythagorean Theorem
1. You can only use the Pythagorean Theorem on a RIGHT triangle (one with a 90° angle). 2. For any triangle, if a2 + b2 = c2 holds true, then that triangle is a RIGHT triangle. 3. It doesn't really matter what leg (side) you label a or b, what matters is that c is the HYPOTENUSE (located directly opposite the 90° angle.)
48 Pythagorean Theorem Worksheet with Answers [Word + PDF] The simplicity of the Pythagorean Theorem worksheet is the best thing about it. What is the Pythagorean Theorem? Formulated in the 6th Century BC by Greek Philosopher and mathematician Pythagoras of Samos, Pythagorean Theorem is a mathematic equation used for a variety of purposes.
Right Triangles - Geometry Pythagorean Theorem Riddle Worksheet This is an 15 question practice worksheet that centers around the concept of the Pythagorean theorem. It requires students to solve for the missing leg or hypotenuse, locate their answer in the solution box to find the corresponding letter to put into the riddle to answer the Pun :)!
Pythagorean Theorem MAZEThis is a self-checking worksheet that allows students to strengthen their skills in finding the missing side (leg or hypotenuse) of a right triangle. Students use their answers to navigate through the puzzle.2. Pythagorean Theorem COLORING ACTIVITY Students are promp Subjects: Math, Math Test Prep, Word Problems Grades:
Ms. Green tells you that a right triangle has a hypotenuse of 13 and a leg of 5. She asks you to find the other leg of the triangle. What is your answer? 12. Two joggers run 8 miles north and then 5 miles west. What is the shortest distance, to the nearest tenth of a mile, they must travel to return to their starting point? 13.
Worksheet #1 - Word Problems Pythagorean Theorem Directions: Solve by drawing a picture, identifying a, b, and c, and applying the Pythagorean Theorem. Do not forget to give your answer with units and show ALL your work to receive full credit. 1. Two sides of a right triangle are 8 inches and 12 inches. a.
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
Question 1: A 9m ladder is placed against a wall. The foot of the ladder is 1.5m from the foot of the wall. How far up the wall does the ladder reach? Question 2: Shown is a square with side length 5cm. Find the length of the diagonal, x. Question 3: Shown is a right angle triangle. Calculate: the perimeter of the triangle.
Pythagorean Theorem S W E N 35 mi 42 mi? 10 in 24 in? 8 ft 15 ft? Solve the word problems. Round the answer to the nearest tenth. Answer Key Level 1: S1 Score : Printable Math Worksheets @ www.mathworksheets4kids.com Name : Mark is on his way home from work. He drives 35 miles due North and then 42 miles due
The Pythagorean Theorem Worksheet is an excellent way to expand your understanding of the Pythagorean Theorem. As you use the worksheet, it is important to remember that 'a' and 'b' represent the shorter lengths on the triangle while c represents the hypotenuse, which is the longest side. Pythagoras Theorem worksheets present you with ...
Detailed Description for All Pythagorean Theorem Worksheets. This Pythagorean Theorem Problems Worksheet will produce problems for practicing solving the lengths of right triangles. You may choose the type of numbers and the sides of the triangle. This worksheet is a great resources for the 6th Grade, 7th Grade, and 8th Grade.
THE PYTHAGOREAN THEOREM PRACTICE PROBLEMS PART A Identify the hypotenuse for each of the following right triangles. c) Determine the area of the indicated square in each of the following diagrams. c) For each of the following, use the given areas to determine side lengths a, b and c. b)
Printable PDF, Google Slides & Easel by TPT Versions are included in this distance learning ready activity which consists of 11 problems involving the Pythagorean Theorem. Students will be required to use the Pythagorean Theorem to solve for the missing leg or hypotenuse of right triangles. ... Geometry Pythagorean Theorem Riddle Worksheet This ...
Add. The sum of the squares of the two shorter sides equals the square of the longest side, so the. triangle is a right triangle. Exercises. Determine whether each set of measures can be the measures of the sides of a. right triangle. Then state whether they form a Pythagorean triple. 1. 30, 40, 50.
1 ACTIVITY: Discovering the Pythagorean Theorem Work with a partner. a. On grid paper, draw any right triangle. Label the lengths of the two shorter sides (the legs) a and b. b. Label the length of the longest side (the hypotenuse) c. c . Draw squares along each of the three sides. Label the areas of the three squares a 2, b 2, and c 2. d.
1 PYTHAGORAS' THEOREM 1 1 Pythagoras' Theorem In this section we will present a geometric proof of the famous theorem of Pythagoras. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other ... This worksheet we will be thinking about a new kind of mathematical proof, called induction. As a ...
Pythagorean theorem is a crucial concept in geometry. It makes it easier to measure the lengths of two smaller sides and calculate units of measurement for any triangle. This theorem explains how in a right-angled triangle, if you square the sides that aren't the hypotenuse (the longest side of any right-angled triangle), then add them ...
The theorem is named after the Greek thinker Pythagoras, born around 570 BC. The theorem states that the area of the square whose side is the hypotenuse of a right angled triangle (the side opposite of the right angle) is equal to the sum of the areas of the squares on the other two sides. Written in algebraic notation this theorem gives the ...
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.
Sheet 1 Determine whether each set of numbers form a Pythagorean triple. 12, 20, 16 1, 7, 5 8, 15, 17 5, 11, 9 29, 20, 21 3 , 14 , 10 24, 18, 30 3, 1, 5 9) 6, 10, 8 10) Determine whether the sides of the triangle form a Pythagorean triple. 2 cm B Printable Math Worksheets @ www.mathworksheets4kids.com Name : Answer Key
These worksheets are available in PDF format to guide the students about basic concepts related to the Pythagorean theorem. Students of KG-3 can download the worksheets and practice the questions from there to gain better knowledge about the topics in the first place. ... In that process, the worksheets for the Pythagorean theorem can help them ...