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GCSE Maths Geometry and Measure

Pythagoras Theorem

Pythagoras’ Theorem

Here we will learn about Pythagoras’ theorem, including how to find sides of a right-angled triangle and using Pythagoras’ theorem to check if a triangle has a right angle or not.

There are also Pythagoras’ theorem worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is Pythagoras’ theorem?

Pythagoras’ theorem is that the square of the longest side of a right angled triangle (called the hypotenuse) is equal to the sum of the squares of the other two sides.

Pythagoras’ theorem is:

\[a^2+b^2=c^2\]

Pythagoras’ Theorem image 1

Side c is known as the hypotenuse.  The hypotenuse is the longest side of a right-angled triangle.  Sides a and b are known as the adjacent sides.  They are adjacent, or next to, the right angle.

We can only use Pythagoras’ theorem with right-angled triangles.

E.g.

Let’s look at this right-angled triangle:

Pythagoras’ Theorem image 2

We can see that three squares have been drawn next to each of the sides of the triangle.

The area of the side of length 3 = 3 \times 3=3^2=9

The area of the side of length 4 = 4\times 4=4^2=16

The area of the side of length 5 = 5\times5=5^2=25

We can see that when we add together the areas of squares on the two shorter sides we get the area of the square on the longest side.

\[9+16=25\]

We can see that when we square the sides of the two shorter sides of a right angled triangle and add them together, we get the square of the longest side.

\[3^2+4^2=5^2\]

3, 4, 5 is known as a Pythagorean triple. 

There are other Pythagorean triples such as 5, 12, 13 and 8, 15, 17 .

If we know two lengths of a right angled triangle, we can use Pythagoras’ theorem to work out the length of the third side.

What is Pythagoras’ theorem?

What is Pythagoras’ theorem?

How to use Pythagoras’ theorem

In order to use Pythagoras’ theorem:

  1. Label the sides of the triangle.
  2. Write down the formula and apply the numbers.
  3. Work out the answer.

Explain how to use Pythagoras’ theorem

Explain how to use Pythagoras’ theorem

Pythagoras’ theorem worksheet

Get your free pythagoras’ theorem worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Pythagoras’ theorem worksheet

Get your free pythagoras’ theorem worksheet of 20+ questions and answers. Includes reasoning and applied questions.

COMING SOON

Pythagoras’ theorem examples

Example 1: finding the length of the hypotenuse (finding the length of the longest side)

Find x and give your answer to 2 decimal places:

Pythagoras’ Theorem example 1 1

  1. Label the sides of the triangle.

It is very important to label the hypotenuse (the longest side) correctly with c .  The adjacent sides, next to the right angle can be labelled a and b either way around as they are interchangeable.

Pythagoras’ Theorem example 1 step 1 1

2Write down the formula and apply the numbers.

\[\begin{aligned} a^2 + b^2 &= c^2 \\\\ 3^2 + 8^2 &= x^2\\\\ x^2 &= 3^2+8^2\\\\ x^2 &= 9 + 64\\\\ x^2 &=73\\\\ x &= \sqrt{73} \end{aligned} \]

3Work out the answer.

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

\[x=\sqrt{73}=8.5440037…\]

The final answer is:

x = 8.54 cm to 2 decimal places

An alternative method is to rearrange the formula and put one calculation into a calculator.

\[\begin{aligned} a^2 + b^2 &= c^2 \\\\ c^2 &=a^2 + b^2 \\\\ c &= \sqrt{a^2 + b^2} \\\\ x &= \sqrt{3^2 + 8^2}\\\\ x &= 8.5440037… \end{aligned} \]

Example 2: finding the length of the hypotenuse (finding the length of the longest side)

Find x and give your answer to 3 significant figures:

Pythagoras’ Theorem example 2 1

It is very important to label the hypotenuse (the longest side) correctly with c .  The adjacent sides, next to the right angle can be labelled a and b either way around as they are interchangeable.


Pythagoras’ Theorem example 2 step 1 1

\[\begin{aligned} a^2 + b^2 &= c^2 \\\\ 7^2 + 9^2 &= x^2\\\\ x^2 &= 7^2 + 9^2\\\\ x^2 &= 49+81\\\\ x^2 &= 130\\\\ x &=\sqrt{130} \end{aligned}\]

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

\[x=\sqrt{130}=11.40175.…\]


The final answer is:


x = 11.4 cm to 3 significant figures


An alternative method is to rearrange the formula and put one calculation into a calculator.

\[\begin{aligned} a^2 + b^2 &= c^2 \\\\ c^2 &=a^2 + b^2 \\\\ c &= \sqrt{a^2 + b^2} \\\\ x &= \sqrt{7^2 + 9^2}\\\\ x &= 11.40175… \end{aligned} \]

Example 3: finding an adjacent side (a short side)

Find x and give your answer to 2 significant figures:

Pythagoras’ Theorem example 3 1

It is very important to label the hypotenuse (the longest side) correctly with c .  The adjacent sides, next to the right angle can be labelled a and b either way around as they are interchangeable.


We will label the short side we are trying to find as side a .


Pythagoras’ Theorem example 3 step 1 1

\[\begin{aligned} a^2 + b^2 &= c^2 \\\\ x^2 + 5^2 &= 8^2\\\\ x^2+25 &= 64 \\\\ x^2 &= 64 – 25 \\\\ x^2 &= 39\\\\ x &=\sqrt{39} \end{aligned}\]

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

\[x=\sqrt{39}=6.244997.…\]


The final answer is:


x = 6.24 cm to 2 significant figures


An alternative method is to rearrange the formula and put one calculation into a calculator.

\[\begin{aligned} a^2 + b^2 &= c^2 \\\\ a^2 &=c^2 – b^2 \\\\ a &= \sqrt{c^2 – b^2} \\\\ x &= \sqrt{8^2 – 5^2}\\\\ x &= 6.244997… \end{aligned} \]

Example 4: finding an adjacent side (a short side)

Find x and give your answer to 3 significant figures:

Pythagoras’ Theorem example 4 1

It is very important to label the hypotenuse (the longest side) correctly with c .  The adjacent sides, next to the right angle can be labelled a and b either way around as they are interchangeable.


We will label the short side we are trying to find as side a .


Pythagoras’ Theorem example 4 step 1 1

\[\begin{aligned} a^2 + b^2 &= c^2 \\\\ x^2 + 11^2 &= 20^2\\\\ x^2+121 &= 400 \\\\ x^2 &= 400 – 121 \\\\ x^2 &= 129\\\\ x &=\sqrt{129} \end{aligned}\]

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

\[x=\sqrt{279}=16.70329.…\]


The final answer is:


x = 16.7 cm to 3 significant figures


An alternative method is to rearrange the formula and put one calculation into a calculator.

\[\begin{aligned} a^2 + b^2 &= c^2 \\\\ a^2 &=c^2 – b^2 \\\\ a &= \sqrt{c^2 – b^2} \\\\ x &= \sqrt{20^2 – 11^2}\\\\ x &= 16.70329… \end{aligned} \]

Example 5: checking if a triangle has a right angle

Is the triangle below a right-angled triangle?

Pythagoras’ Theorem example 5 1

It is very important to label the hypotenuse (the longest side) correctly with c .  The adjacent sides, next to the right angle can be labelled a and b either way around as they are interchangeable.


Pythagoras’ Theorem example 5 step 1 1

\[\begin{aligned} a^2 + b^2 &= c^2 \\\\ 8^2 + 10^2 &= 13^2\\\\ 64+100 &= 169 \\\\ 164 &= 169 \end{aligned}\]


But this is NOT correct.  Pythagoras’ theorem only works with right-angled triangles.

Because

\[8^2 + 10^2 \neq 13^2\]


The sides of the triangles do not fit with Pythagoras’ theorem.  Therefore the triangle is NOT a right-angled triangle.

Example 6: checking if a triangle has a right angle

Is the triangle below a right-angled triangle?

Pythagoras’ Theorem example 6 1

It is very important to label the hypotenuse (the longest side) correctly with c .  The adjacent sides, next to the right angle can be labelled a and b either way around as they are interchangeable.


Pythagoras’ Theorem example 6 step 1 1

\[\begin{aligned} a^2 + b^2 &= c^2 \\\\ 6^2 + 8^2 &= 10^2\\\\ 36+64 &= 100 \\\\ 100 &= 100 \end{aligned}\]


This is correct.  Pythagoras’ theorem only works with right-angled triangles.

Because

\[6^2 + 8^2 = 10^2\]


The sides of the triangles fit with Pythagoras’ theorem.  Therefore the triangle is a right-angled triangle.

Common misconceptions

  • Make sure you identify the hypotenuse

It is very important to make sure that the hypotenuse is correctly identified and labelled c .

  • Right-angled triangles may be in different orientations

The triangles can be drawn in different orientations.  These are right-angled triangles in different orientations

Pythagoras’ Theorem common misconception 1 1
  • Make sure you write what your calculator shows, before rounding

It is very easy to make a mistake at the end with rounding.  So make sure you have shown your working out.

  • Lengths of sides do not have to be whole numbers

Lengths can be decimals, fractions or even irrational numbers such as surds,

e.g. \sqrt{2} .

  • Pythagoras’ theorem only works on right-angled triangles

Pythagoras’ theorem only works on right-angle triangles, but with a bit of thought it can be used on other triangles.  For example an isosceles triangle can be made into 2 right-angled triangles by putting in the line of symmetry.

Pythagoras’ Theorem common misconception 2 1
  • Pythagoras’ theorem can also be applied to other shapes

An application of Pythagoras’ theorem is to extend it to work on other shapes such as a trapezium.

Pythagoras’ Theorem common misconception 3 1

  • Do not round too early

If you need to use Pythagoras’ theorem in a question with multiple steps, do not round until the very end of the question or you will lose accuracy.  For example, you may need to find the height of a triangle, and then use that height to find its area.

Practice Pythagoras’ theorem questions

1. Find side x. Give your answer to 2 decimal places:

 

Practice Pythagoras’ Theorem questions 1 1

8.60 cm
GCSE Quiz True

8.6 cm
GCSE Quiz False

4.90 cm
GCSE Quiz False

4.9 cm
GCSE Quiz False

Practice Pythagoras’ Theorem questions 1 explanation 1

 

\begin{aligned} a^2 + b^2 &= c^2 \\\\ 7^2 + 5^2 &= x^2 \\\\ x^2 &= 7^2 + 5^2\\\\ x^2 &= 49+25 \\\\ x^2 &= 74\\\\ x &= \sqrt{74}\\\\ x &= 8.602325… \end{aligned}

 

x=8.60 cm to 2 decimal places

2. Find side x. Give your answer to 3 decimal places:

 

Practice Pythagoras’ Theorem questions2 1

17.2 cm
GCSE Quiz True

17.20 cm
GCSE Quiz False

9.79 cm
GCSE Quiz False

9.80 cm
GCSE Quiz False

Practice Pythagoras’ Theorem questions 2 explanation 1

 

\begin{aligned} a^2 + b^2 &= c^2 \\\\ 14^2 + 10^2 &= x^2\\\\ x^2 &= 14^2 + 10^2\\\\ x^2 &= 196+10\\\\ x^2 &= 296\\\\ x &= \sqrt{296}\\\\ x &= 17.20465… \end{aligned}

 

x=17.2 cm to 3 decimal places

3. Find side x. Give your answer to 2 decimal places:

 

Practice Pythagoras’ Theorem questions 3 1

14.97 cm
GCSE Quiz True

14.96 cm
GCSE Quiz False

20.59 cm
GCSE Quiz False

2.58 cm
GCSE Quiz False

Practice Pythagoras’ Theorem questions 3 explanation 1

 

\begin{aligned} a^2 + b^2 &= c^2 \\\\ x^2 + 10^2 &= 18^2\\\\ x^2+100 &= 324 \\\\ x^2 &= 324 – 100 \\\\ x^2 &= 224\\\\ x &=\sqrt{224}\\\\ x &= 14.96662… \end{aligned}

 

x=14.97 cm to 2 decimal places

4. Find side x. Give your answer to 2 decimal places:

 

Practice Pythagoras’ Theorem questions 4 1

6.75 cm
GCSE Quiz True

6.76 cm
GCSE Quiz False

7.62 cm
GCSE Quiz False

7.63 cm
GCSE Quiz False

Practice Pythagoras’ Theorem questions 4 explanation 1

 

\begin{aligned} a^2 + b^2 &= c^2 \\\\ x^2 + 2.5^2 &= 7.2^2\\\\ x^2+6.25 &= 51.84 \\\\ x^2 &= 51.84 – 6.25 \\\\ x^2 &= 45.59\\\\ x &=\sqrt{45.59}\\\\ x &= 6.75203… \end{aligned}

 

x=6.75 cm to 2 decimal places

5. Say if this a right-angled triangle and give your reason:

 

Practice Pythagoras’ Theorem questions 5 1

Yes because 12^2+5^2=13^2

GCSE Quiz True

No because 12^2+5^2 13^2

GCSE Quiz False

Yes because I measured the angle and it was 90^{\circ}

GCSE Quiz False

No because I measured the angle and it was not 90^{\circ}

GCSE Quiz False

Practice Pythagoras’ Theorem questions 5 explanation 1

 

\begin{aligned} a^2 + b^2 &= c^2 \\\\ 12^2 + 5^2 &= 13^2\\\\ 144+25 &= 169 \\\\ 169 &= 169 \end{aligned}

 

This is correct. Pythagoras’ Theorem only works with right-angled triangles.

 

Therefore the triangle is a right-angled triangle.

6. Say if this a right-angled triangle and give your reason:

 

Practice Pythagoras’ Theorem questions 6 1

No because 6^2+13^2 14^2

GCSE Quiz True

Yes because 6^2+13^2=14^2

GCSE Quiz False

Yes because I measured the angle and it was 90^{\circ}

GCSE Quiz False

No because I measured the angle and it was not 90^{\circ}

GCSE Quiz False

Practice Pythagoras’ Theorem questions 6 explanation 1

 

\begin{aligned} a^2 + b^2 &= c^2 \\\\ 6^2 + 13^2 &= 14^2\\\\ 36+169 &= 196 \\\\ 205 &= 169 \end{aligned}

 

This is NOT correct. Pythagoras’ Theorem only works with right-angled triangles.

 

Therefore the triangle is NOT a right-angled triangle.

Pythagoras’ theorem GCSE questions

1. ABC is a right-angled triangle.

 

Pythagoras’ Theorem GCSE questions 1 1

 

Calculate the length of AC.

 

Give your answer correct to 3 significant figures.

 

(3 marks)

Show answer
7.3^2 + 12.7 ^2 = 214.58

(1)

 

\sqrt{214.58}

(1)

 

AC = 14.6485… = 14.6 cm

(1)

2. Triangle ABC has a perimeter of 19 cm.

 

AB = 5 cm \\ BC = 6 cm

 

By calculation, deduce whether triangle ABC is a right-angled triangle.

 

(4 marks)

Show answer
19-(5+6)=8

 

Third side is 8 cm and is the longest so it is the hypotenuse

(1)

 

5^2 + 6 ^2 = 61

(1)

 

\sqrt{61}=7.81…

 

OR

 

8^2=64

(1)

 

Triangle ABC is NOT a right-angled triangle

(1)

3. A frame is made from wire

 

Pythagoras’ Theorem GCSE questions 3 1

 

The frame is in the shape of a rectangle 10 cm by 15 cm.

 

The diagonal of the rectangle is also made from wire.

 

Calculate the total length of wire needed to make the frame and the diagonals.

 

Give your answer correct to 1 decimal place.

(4 marks)

Show answer
10^2 + 15 ^2 = 325

(1)

 

\sqrt{325}=18.02775

(1)

 

18.02775.. + (2\times 15) + (2\times 10)

(1)

 

68.027756…. = 68.0 cm

(1)

Did you know?

Pythagoras’ theorem is named after a Greek mathematician who lived about 2500 years ago, however the ancient Babylonians used this rule about 4 thousand years ago!  At the same time the Egyptians were using the theorem to help them with right angles when building structures.

Learning checklist

You have now learned how to:

  • Use Pythagoras’ theorem to find the length of the longest side of a right-angled triangle – its hypotenuse
  • Use Pythagoras’ theorem to find the length of one of the shorter sides of a right-angled triangle

The next lessons are

  • Trigonometry – using sine, cosine and tangent ratios
  • Coordinate geometry – the distance between two points
  • 3D Pythagoras

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