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Pythagoras Theorem Questions

Welcome to our Pythagoras' Theorem Questions area. Here you will find help, support and questions to help you master Pythagoras' Theorem and apply it.

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Pythagoras' Theorem Questions

Here you will find our support page to help you learn to use and apply Pythagoras' theorem.

Please note: Pythagoras' Theorem is also called the Pythagorean Theorem

There are a range of sheets involving finding missing sides of right triangles, testing right triangles and solving word problems using Pythagoras' theorem.

Using these sheets will help your child to:

• learn Pythagoras' right triangle theorem;
• use and apply the theorem in a range of contexts to solve problems.

Pythagoras' Theorem

Pythagoras' Theorem - in more detail

Pythagoras' theorem states that in a right triangle (or right-angled triangle) the sum of the squares of the two smaller sides of the triangle is equal to the square of the hypotenuse.

In other words, \[ a^2 + b^2 = c^2 \]

where c is the hypotenuse (the longest side) and a and b are the other sides of the right triangle.

What does this mean?

This means that for any right triangle, the orange square (which is the square made using the longest side) has the same area as the other two blue squares added together.

Other formulas that can be deduced from the Pythagorean theorem

As a result of the formula \[ a^2 + b^2 = c^2 \] we can also deduce that:

• \[ b^2 = c^2 - a^2 \]
• \[ a^2 = c^2 - b^2 \]
• \[ c = \sqrt{a^2 + b^2} \]
• \[ b = \sqrt {c^2 - a^2} \]
• \[ a = \sqrt {c^2 - b^2} \]

Pythagarean Theorem Examples

Example 1) find the length of the missing side..

In this example, we need to find the hypotenuse (longest side of a right triangle).

So using pythagoras, the sum of the two smaller squares is equal to the square of the hypotenuse.

This gives us \[ 4^2 + 6^2 = ?^2 \]

So \[ ?^2 = 16 + 36 = 52 \]

This gives us \[ ? = \sqrt {52} = 7.21 \; cm \; to \; 2 \; decimal \; places \]

Example 2) Find the length of the missing side.

In this example, we need to find the length of the base of the triangle, given the other two sides.

This gives us \[ ?^2 + 5^2 = 8^2 \]

So \[ ?^2 = 8^2 - 5^2 = 64 - 25 = 39 \]

This gives us \[ ? = \sqrt {39} = 6.25 \; cm \; to \; 2 \; decimal \; places \]

Pythagoras' Theorem Question Worksheets

The following questions involve using Pythagoras' theorem to find the missing side of a right triangle.

The first sheet involves finding the hypotenuse only.

A range of different measurement units have been used in the triangles, which are not drawn to scale.

• Pythagoras Questions Sheet 1
• PDF version
• Pythagoras Questions Sheet 2
• Pythagoras Questions Sheet 3
• Pythagoras Questions Sheet 4

Pythagoras' Theorem Questions - Testing Right Triangles

The following questions involve using Pythagoras' theorem to find out whether or not a triangle is a right triangle, (whether the triangle has a right angle).

If Pythagoras' theorem is true for the triangle, and c 2 = a 2 + b 2 then the triangle is a right triangle.

If Pythagoras' theorem is false for the triangle, and c 2 = a 2 + b 2 then the triangle is not a right triangle.

• Pythagoras Triangle Test Sheet 1
• Pythagoras Triangle Test Sheet 2

Pythagoras' Theorem Questions - Word Problems

The following questions involve using Pythagoras' theorem to solve a range of word problems involving 'real-life' type questions.

On the first sheet, only the hypotenuse needs to be found, given the measurements of the other sides.

Illustrations have been provided to support students solving these word problems.

• Pythagoras Theorem Word Problems 1
• Pythagoras Theorem Word Problems 2

Geometry Formulas

• Geometry Formula Sheet

Here you will find a support page packed with a range of geometric formula.

Included in this page are formula for:

• areas and volumes of 2d and 3d shapes
• interior angles of polygons
• angles of 2d shapes
• triangle formulas and theorems

This page will provide a useful reference for anyone needing a geometric formula.

Triangle Formulas

Here you will find a support page to help you understand some of the special features that triangles have, particularly right triangles.

Using this support page will help you to:

• understand the different types and properties of triangles;
• understand how to find the area of a triangle;
• know and use Pythagoras' Theorem.

All the free printable geometry worksheets in this section support the Elementary Math Benchmarks.

• Geometry Formulas Triangles

Here you will find a range of geometry cheat sheets to help you answer a range of geometry questions.

The sheets contain information about angles, types and properties of 2d and 3d shapes, and also common formulas associated with 2d and 3d shapes.

Included in this page are:

• images of common 2d and 3d shapes;
• properties of 2d and 3d shapes;
• formulas involving 2d shapes, such as area and perimeter, pythagoras' theorem, trigonometry laws, etc;
• formulas involving 3d shapes about volume and surface area.

Using the sheets in this section will help you understand and answer a range of geometry questions.

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Pythagorean Theorem Exercises

Pythagorean theorem practice problems with answers.

There are eight (8) problems here about the Pythagorean Theorem for you to work on. When you do something a lot, you get better at it. Let’s get started!

Here’s the Pythagorean Theorem formula for your quick reference.

Note: drawings not to scale

Problem 1: Find the value of [latex]x[/latex] in the right triangle.

[latex]10[/latex]

Problem 2: Find the value of [latex]x[/latex] in the right triangle.

[latex]4\sqrt 7 [/latex]

Problem 3: Find the value of [latex]x[/latex] in the right triangle.

[latex]3[/latex]

Problem 4: The legs of a right triangle are [latex]5[/latex] and [latex]12[/latex]. What is the length of the hypotenuse?

[latex]13[/latex]

Problem 5: The leg of a right triangle is [latex]8[/latex] and its hypotenuse is [latex]17[/latex]. What is the measure of its other leg?

[latex]15[/latex]

Problem 6: Suppose the shorter leg of a right triangle is [latex]\sqrt 2[/latex]. The longer leg is twice the shorter leg. Find the hypotenuse.

[latex]\sqrt {10} [/latex]

Problem 7: The hypotenuse of a right triangle is [latex]4\sqrt 2[/latex]. If the longest leg is half the hypotenuse, what is the length of the shortest leg?

[latex]2\sqrt 6[/latex]

Problem 8: Find the value of [latex]x[/latex] of the right isosceles triangle.

[latex]\sqrt 6[/latex]

You might also like these tutorials:

• Pythagorean Theorem
• Pythagorean Triples
• Generating Pythagorean Triples

• Consumer Math
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Fun Pythagorean Theorem Activities and Teaching Ideas

4 comments:

Thank you. This is great.

Thank you for all your hard work in providing resources for math teachers. I use a lot of these for scaffolding and differentiating for my high school students who are at the pre-algebra instructional level.

It's really my pleasure, Ms. Turner. Thank you for reading my blog and leaving a comment. I know how tough your job is, so thank you also for all your hard work. I hope you and your students are having a great year.

Thank you sm!! ur so helpfull~

Ollie Lovell

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Pythagoras’ Theorem: My attempt to write challenging questions and engage students

On my last placement I taught a unit on Pythagoras' theorem to my year 9 class. I found it deceptively difficult. Why? The challenge was twofold: First, students had seen the theorem in previous years so it was a bit old hat to them. Second, I wondered how to stretch students and provide them with variety and challenge beyond the old trick of calling the hypotenuse a ladder?

I was thinking to myself ‘ WHY does Pythagoras’ theorem actually matter, and how can I make that tangible and relevant to students? I came up with the following slideshow.

I had this first slide on the screen as students walked into the classroom, it created a bit of a buzz in the room…

I got students to vote (hands up) on which corner they thought was square (which do you think it is?), then revealed the answer.

I've presented the remainder of the slideshow in the following video with examples of the kind of teacher questions and the script that I used. Seemed easier and clearer than writing it all out…

(If you want to use any of this you can download the slideshow file here:  Pythagoras Slides_Oliver Lovell_www.ollielovell.com )

Clarifying points…

• I used the qualifier of ‘on the macro scale' when talking about ‘no squares in nature' because I thought it's probably possible for there to be perfect squares in crystalline atomic structures or something like that.
• Perhaps an oversight in the video is a discussion of the meaning of the word ‘breadth' (I've become more aware of the vocab that I'm using in class since my resent teaching placement in Myanmar where the students had very limited english. Post to come).

The general structure of lessons in this unit from then on was a bit of book work (using the ‘points' system that I'll write about in another post) with a challenge question for those students who got all of their points done. Lessons were wrapped up with students sharing on the board their approach to the challenge question and what they had learned from attempting it. I also book-ended most lessons with a micro-revision, 3 questions on the basics as a mini-test at the start, and an exit card at the end to help me to keep track of how students were progressing.

Below are the questions that I used as challenge questions. I wrote all of the following questions, excepting the final question, which I found here . I'd also like to acknowledge Dr Max Stephens who suggested that I create questions that encourage students to solve for exact values rather than untidy decimals. I've left out answers on purpose in the knowledge that future students will possibly visit this post.

Pythagoras Extension Questions:

You’ve always got to include a ladder question don’t you? (this one is a little bit different).

A Cat is stuck in a tall tree and needs to be saved by the Firefighters. The fire truck has a 35m ladder. The closest that it can get to the tree (because there are parked cars in the way) is 12m. The cat is in the tree, 39m up. Comment on the likelihood of the cat getting saved and justify your suggestion mathematically.

Q:  A cube has side length of 10cm. what is the length of the diagonal through its centre? (could accompany this with the picture below).

Alternatively you could ask, Q:  ‘What is the length of d in the following if the side lengths are 10cm?' (referring to the above image)

Prior to giving students the above question we’d had a discussion about the formula of the diagonal of a box and students had worked out that it’s just the square root of the sum of the squares of the box’s dimensions. This helped them to solve this question. One of the students actually proposed this ‘pythagoras in 3 dimensions' formula, so I encouraged them all to check it and see if the proposal seemed plausible.

An additional question that I also posed to students was,  Q:  “Show that the formula for the diagonal of a box is √a²+b²+c².” A discussion of this also helped to scaffold the students.

Q:  A rectangular prism has sides of length a, 4a/3, and 4a. It has an internal diagonal of 26cm. What are the dimensions of the rectangular prism?

Or, you could word it in a more tricky way:   Q:  A rectangular prism has a width one quarter of its length and a height one third of its width. It has an internal diagonal of 26cm. What are the dimensions of the rectangular prism?

Some other interesting shapes…

Another question:

In the 6 sided pyramid pictured below, what is the height of the apex ‘V’ above the point  ‘O’ ? (You can assume that O is in the centre of the base, the base is a regular hexagon, and V is directly above O).

Resources to run a PD on ‘Hinge Questions’: Based on Dylan Wiliam’s ‘ACER Teacher’ podcast

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Mr Barton Maths Podcast

Long-form conversations about teaching and learning with craig barton.

Pythagoras and Trigonometry Collection

Interesting teaching ideas for introducing and revising pythagoras’ theorem and trigonometry.

Pythagoras’ theorem and trigonometry are two of those classic topics that pupils revisit year-on-year. This is partly because these topics come in many forms and interesting contexts, from basic Pythagoras and Soh-Cah-Toa, to graphs of trigonometric functions and calculus.

With the increased emphasis on ratio in the new GCSE specification, trigonometry is likely to start appearing in non-calculator papers as well. As such, it is more important than ever for students to get to grips with this crucial and fascinating area of mathematics.

Fortunately, TES Maths is here to help!

Below is a selection of 15 top Pythagoras and trigonometry resources, all uploaded by the talented members of the TES Maths community. Why not use them to give you some new ideas for teaching these topics across all year groups?

Craig Barton, TES Maths adviser

• Perigal’s dissection Encourage pupils to stumble upon Pythagoras’ theorem by themselves by first introducing them to Perigal’s cut-and-shift proof.
• Introduction to Pythagoras’ theorem This well-differentiated lesson neatly introduces the famous theorem and the correct use of the calculator in solving Pythagoras-related problems.
• Interactive Pythagoras lesson Culminating in a treasure hunt, this action-packed lesson starts with an element of discovery before progressing on to using Pythagoras in real-life contexts.
• Have I got hypotenuse for you? Revise the key aspects of Pythagoras in a fun and effective way with this select-and-reveal starter and plenary activity.
• Trigonometry investigation Give your students the opportunity to discover the relationship between trigonometry and ratio in this fully-resourced investigation.
• Introduction to trigonometry Introduce sin, cos and tan for the first time with this clear, well-structured lesson, which can be easily adapted to suit your class’ needs.
• Soh-Cah-Toa Crammed full of differentiated activities, this complete lesson on trigonometry uses Bloom’s taxonomy questioning to challenge pupils.
• Batman-themed word problems Help Gotham City’s Police Commissioner catch criminals while recapping existing learning on trigonometry with these engaging problems.
• Pirate problem-solving Revise trigonometry, Pythagoras’ theorem and even geometry in this themed activity, in which pupils are tasked with finding Admiral Angle’s buried treasure.
• Pythagoras or trigonometry? Clarify the process of deciding whether a question requires the use of Pythagoras or Soh-Cah-Toa with this handy flow chart.
• 3D trigonometry Stretch more able learners with this innovative task, which is ideal for pair work and contains plenty of opportunities for peer and self-assessment.
• Finding exact trigonometric values Use these clear worksheets to encourage students to become familiar with ‘special triangles’, in order to be able to calculate exact trigonometric values without a calculator.
• Sine and cosine rule lesson This step-by-step lesson plan with precise explanations and well-worked examples is ideal for KS4 classes.
• Sine and cosine sorting exercise Put your pupils’ understanding of when to use the sine and cosine rule to the test with this versatile resource, with plenty of AfL opportunities.
• Pythagoras and trigonometry revision booklet Containing lots of exam-style questions, this comprehensive workbook has plenty of space for students to show their working and explain their reasoning.

2 thoughts on “ Pythagoras and Trigonometry Collection ”

Hardly unique! But thanks for sharing. Your website is a really useful collection of resources.

Good point. That was the TES editor’s choice of title. I have changed to “interesting”!

Leave a Reply

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GCSE Pythagoras Theorem Questions and Answers

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1. What is Pythagoras Theorem

Pythagoras theorem states that the square of the Hypotenuse is equal to the sum of the squares of the other two sides. 

If the sides of the right-angled triangle are labelled a, b and c then Pythagoras' theorem states : a ² + b ² = c ²

In the above diagram, side c is known as the hypotenuse, the longest side of a right-angled triangle and is opposite the right angle. Side “a” and side “b” are known as the adjacent sides.

Note: The hypotenuse is always the longest side of the right angled triangle.

If any two sides of a right angled triangle are known, one can use Pythagoras theorem to work out the length of the third side.

1.1 How to use Pythagoras Theorem

In order to use Pythagoras theorem, first label the sides of the triangle. Next write down the formula and apply the numbers and finally work out the answer.

In order to use Pythagoras theorem to find hypotenuse for the given triangle:

First label sides as a = 7 cm, b = 24 cm, c = c

It is very important to label the hypotenuse correctly with c. The adjacent sides, next to the right angle can be labelled a and b according to the size.

Next write down the formula a ² + b ² = c ²  

Apply the numbers and finally work out the answer:

Make sure you give your final answer in the correct form including units where appropriate.

1.2 Example Questions on Pythagoras theorem

Example 1: On finding the length of the hypotenuse 

Find x and give your answer to 4 significant figures:

Solution : 

Example 2: on finding an adjacent side.

Find x in the given triangle.

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A grade of 4 or 5 would be considered "good" because the government has established a 4 as the passing grade; a grade of 5 is seen as a strong pass. Therefore, anything that exceeds this level would be considered good. You can practice GCSE Maths topic-wise questions to score good grades in the GCSE Maths exam.

You can get a high score in GCSE Maths through meticulous practice of GCSE Maths topic-wise questions and GCSE Maths past papers .

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Pythagoras' theorem – complete topic booklet

A useful booklet that works through Pythagoras' theorem from identifying right-angled triangles to 3D Pythagoras, and includes problem-solving practice. Objectives are:

• identify and label right-angled triangles
• explain Pythagoras' theorem
• calculate a hypotenuse
• calculate a shorter side
• determine whether a triangle has a right angle
• exact answers with surds
• areas and perimeters
• worded problems
• 3D Pythagoras

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Know when to use Pythagoras or Trigonometry to solve problems

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Lesson details

Key learning points.

• In this lesson, we will recognise when it is appropriate to use Pythagoras or Trigonometry when finding missing lengths and angles in right angle triangles.

This content is made available by Oak National Academy Limited and its partners and licensed under Oak’s terms & conditions (Collection 1), except where otherwise stated.

Starter quiz

3 questions.

Lesson appears in

Unit maths / trigonometry 3.

Resourceaholic

Ideas and resources for teaching secondary school mathematics

• Blog Archive

2 September 2014

Pythagoras' theorem.

11 comments:

Great post, some lovely questions and resources here I've not seen before. On the subject of triples, you may find this interesting: http://wp.me/p2z9Lp-bP And my very first ever (albeit not great) blog post was about Pythagoras in 3d! http://wp.me/p2z9Lp-9

Thanks Cav. Aah, your first post from 2012! I bet that feels like a long time ago. Love your triples post, I'll definitely use that method for finding triples. Also the proof by induction for FP1, great stuff.

Here’s a lesson I did when I visited a teacher with a year 8 class in a middle school in the summer term. She asked me if I could do a lesson which tied in some assorted revision and practice while leading the way towards Pythagoras – without spoiling the fun for their teachers in secondary school. Phew! I asked for five minutes to think about it and this is what we did. 1…..I asked the children to construct a rectangle as perfect as they could make it. (Lots of revision of basic 2-D shapes and their properties, vocabulary, and of course accurate use of instruments.) 2…..I then asked them to draw a diagonal of their triangle and rub out one of the two triangles. 3…..I asked them to construct an equilateral triangle on each side of their triangle and to find the area of each of the three triangles. (So there was lots more practice in constructing triangles and using equipment, not to mention finding an assortment of methods to find the areas.) 4…..Finally, I asked them to post their three answers on the board and see what they could observe. I was pretty pleased that I'd managed to come up with the whole lesson quickly from scratch. The word Pythagoras was never mentioned, but I rather hoped the lesson would have to come mind when they met one of the standard diagrams in a few months’ time. Of course, constructing semi-circles on the sides would have worked just as well, but I thought there’d be more mileage in using equilateral triangles.

Thanks for sharing this lesson Alan, it's fantastic. I love the use of accurate constructions, a skill that's always worth practising. I'm certainly going to borrow these ideas.

Two animations that might be of interest to readers of this post: http://www.mathimation.co.uk/triangles/pythagoras-theorem-proof/ and http://burymathstutor.co.uk/Pythagoras.html

Interesting extensions of Pythagoras theorem. https://www.youtube.com/watch?v=YRdKI71tx-4 Bet you didn't know this about Pythagoras. https://www.youtube.com/watch?v=li8g0FMD3wc

Lovely, thank you!

Hi Jo, Just popping by for some inspiration! So many great ideas for my year 9s! A great activity I've done for Pythagoras is giving each half of the class a rope with 13 knots in at equal distances and then ask them to make me a right angled triangle. It works as a great little race for students to start problem solving. Dan

Thanks Dan, fantastic idea!

I'm not maths teacher (or a physics teacher) but a former engineer turned trade union official. I still love maths and physics though and during my attempt to get a decent understanding of special relativity I came across this very simple way of understanding special relativity and time dilatation using only Pythagoras. There is, of course, more complex maths involved in relativity, but this use of only Pythagoras is brilliant and could be used with older students or 6th formers to get a conceptual understanding of an important physics concept. I wish someone had shown this to me when I was at school as I may have studied physics instead of going to Uni to chemistry and dropping out. Is worth checking out http://www.emc2-explained.info/The-Light-Clock/#.VgBs6ZcYNOI

Great post and ideas, thanks. Been inspired to post a link to an interactive Pythagoras tool of our own: http://www.mathelize.co.uk/pt.html

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GCSE Higher Revision - 6.3. Applying Pythagoras' Theorem in Real-Life Situations.

Subject: Whole school

Age range: 11-14

Resource type: Other

Last updated

21 March 2021

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This resource is for GCSE Higher revision but can be used for KS3. This lesson is about being able to solve real-life problems involving Pythagoras’ Theorem. There are many examples and questions for students to work through on Grade C Pythagoras problems. Also, depending on the ability of your group, there is an extension on using Pythagoras to calculate the area of triangles with some Grade B questions on this.

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salhuskisson

thanks for sharing

Empty reply does not make any sense for the end user

Very good questions, but could do with an answer sheet.

thanks, great work, my students will love this

maurisalee_vanas

Thanks! This is great!

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Word problems on Pythagorean Theorem

Learn how to solve different types of word problems on Pythagorean Theorem .

Pythagoras Theorem can be used to solve the problems step-by-step when we know the length of two sides of a right angled triangle and we need to get the length of the third side.

Three cases of word problems on Pythagorean Theorem :

Case 1: To find the hypotenuse where perpendicular and base are given.

Case 2: To find the base where perpendicular and hypotenuse are given.

Case 3: To find the perpendicular where base and hypotenuse are given.

Word problems using the Pythagorean Theorem:

1. A person has to walk 100 m to go from position X in the north of east direction to the position B and then to the west of Y to reach finally at position Z. The position Z is situated at the north of X and at a distance of 60 m from X. Find the distance between X and Y.

⇒ 200x = 10000 + 3600

⇒ 200x = 13600

⇒ x = 13600/200

Therefore, distance between X and Y = 68 meters.

Therefore, length of each side is 8 cm.

Using the formula solve more word problems on Pythagorean Theorem.

3. Find the perimeter of a rectangle whose length is 150 m and the diagonal is 170 m.

In a rectangle, each angle measures 90°.

Therefore PSR is right angled at S

Using Pythagoras theorem, we get

⇒ PS = √6400

Therefore perimeter of the rectangle PQRS = 2 (length + width)

                                                          = 2 (150 + 80) m

                                                          = 2 (230) m

                                                          = 460 m

4. A ladder 13 m long is placed on the ground in such a way that it touches the top of a vertical wall 12 m high. Find the distance of the foot of the ladder from the bottom of the wall.

Let the required distance be x meters. Here, the ladder, the wall and the ground from a right-angled triangle. The ladder is the hypotenuse of that triangle.

According to Pythagorean Theorem,

Therefore, distance of the foot of the ladder from the bottom of the wall = 5 meters.

5. The height of two building is 34 m and 29 m respectively. If the distance between the two building is 12 m, find the distance between their tops.

The vertical buildings AB and CD are 34 m and 29 m respectively.

Draw DE ┴ AB

Then AE = AB – EB but EB = BC

Therefore AE = 34 m - 29 m = 5 m

Now, AED is right angled triangle and right angled at E.

⇒ AD = √169

Therefore the distance between their tops = 13 m.

The examples will help us to solve various types of word problems on Pythagorean Theorem.

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