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15 Pythagorean Theorem Practice Problems For 8th Grade

Pythagorean Theorem practice problems involve using the relationship between the sides of a right triangle to calculate missing side lengths in triangles. The Pythagorean Theorem is introduced in 8th grade and is used to solve a variety of problems across high school.

Here, you’ll find a selection of Pythagorean Theorem questions that demonstrate the different types of questions students are likely to encounter in 8th grade.

What is the Pythagorean Theorem?

The Pythagorean Theorem is the geometric theorem that states that the square of the hypotenuse (longest side) of a right triangle is equal to the sum of the squares of the two shorter sides of the triangle.

This can be written as a^2+b^2=c^2 for a triangle labeled like this:

15 Pythagoras Theorem image 1
15 Pythagoras Theorem Practice Problems

15 Pythagoras Theorem Practice Problems

Wish you could have the 15 Pythagoras Theorem questions from this blog in a ready-to-go worksheet? We've done just that!

How to answer Pythagorean Theorem questions

1 – Label the sides of the triangle a, b, and c.
Note that the hypotenuse, the longest side of a right triangle, is opposite the right angle and will always be labeled.

15 Pythagoras Theorem image 2

2 – Write down the formula and substitute the values>

a^2+b^2=c^2

3 – Calculate the answer.
You may be asked to give your answer in an exact form or round to a given degree of accuracy, such as a certain number of decimal places or significant figures.

Pythagorean Theorem in real life

Pythagorean Theorem has many real-life uses, including in architecture and construction, navigation and surveying.

Pythagorean Theorem in 8th grade

Pythagorean Theorem is usually introduced in middle school, as it is a part of the 8th grade Common Core Math Standards.

The emphasis in middle school is on students being able to:

  • Explain the Pythagorean Theorem;
  • Use the theorem to solve mathematical and real-world problems – with both 2D and 3D figures;
  • Use the theorem to calculate the distance between two points on a coordinate grid.

The process for solving any Pythagoras Theorem problem always begins by identifying the relevant right-angled triangle and labeling the sides a, b, c. If there is not a diagram in the question, it can be helpful to draw one.

Where necessary, round your answers to 3 significant figures.

Pythagorean Theorem practice problems

1. A ship sails 6 \, km East and then 8 \, km North. Find the ship’s distance from its starting point.

 

15 Pythagoras Theorem question 1

14 \, km
GCSE Quiz False

10 \, km
GCSE Quiz True

5.29 \, km
GCSE Quiz False

2 \, km
GCSE Quiz False

15 Pythagoras Theorem answer 1

 

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

 

The ship is 10 kilometers from its starting point.

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

The ship is 10 kilometers from its starting point.

2. A ladder is 5 \, m long. The base of the ladder is 3 \, m from the base of a vertical wall. How far up the wall does the ladder reach?

 

15 Pythagoras Theorem question 2

8 \, m
GCSE Quiz False

5.83 \, m
GCSE Quiz False

4 \, m
GCSE Quiz True

16 \, m
GCSE Quiz False

15 Pythagoras Theorem answer 2

 

\begin{aligned} b^{2}&=c^{2}-a^{2}\\ b^{2}&=5^{2}-3^{2}\\ b^{2}&=25-9\\ b^{2}&=16\\ b&=\sqrt{16}\\ b&=4 \, m \end{aligned}

The ladder reaches 4 meters up the wall.

3. Alex and Sam start from the same point. Alex walks 400 meters west. Sam walks x meters south, until they are 600 \, m apart from each other. How far does Sam walk?

 

200 \, m
GCSE Quiz False

447 \, m
GCSE Quiz True

721 \, m
GCSE Quiz False

1000 \, m
GCSE Quiz False

15 Pythagoras Theorem answer 3

 

\begin{aligned} a^{2}&=c^{2}-b^{2}\\ x^{2}&=600^{2}-400^{2}\\ x^{2}&=360000-160000\\ x^{2}&=200000\\ x&=\sqrt{200000}\\ x&=447.2135955\\ x&=447 \, m ~\text{(3sf)} \end{aligned}

4. A television’s size is the measurement from the upper left hand corner of the television to the bottom right hand corner. Find the size of this television.

 

15 Pythagoras Theorem question 4

75 inches

GCSE Quiz False

150 inches

GCSE Quiz False

39.7 inches

GCSE Quiz False

55.1 inches

GCSE Quiz True

15 Pythagoras Theorem answer 4

 

\begin{aligned} a^2+b^2&=c^2\\ 48^2+27^2&=c^2\\ 2304+729&=c^2\\ 3033&=c^2\\ c&=\sqrt{3033}\\ c&=55.07267925\\ c&=55.1 \, \mathrm{inches} ~ \text{(3sf)} \end{aligned}

5. The pole of a sailing boat is supported by a rope from the top of the pole to an anchor point on the deck. The pole is 4 \, m long and the rope is 4.5 \, m long. Calculate the distance from the base of the pole to the anchor point of the rope on the deck.

 

15 Pythagoras Theorem question 5

2.06 \, m
GCSE Quiz True

6.02 \, m
GCSE Quiz False

8.5 \, m
GCSE Quiz False

0.5 \, m
GCSE Quiz False

15 Pythagoras Theorem answer 5

 

\begin{aligned} b^{2}&=c^{2}-a^{2}\\ b^{2}&=4.5^{2}-4^{2}\\ b^{2}&=20.25-16\\ b^{2}&=4.25\\ b&=\sqrt{4.25}\\ b&=2.061552813\\ b&=2.06 \, m ~\text{(3sf)} \end{aligned}

6. Work out the length of the diagonal of a square with 8 \, cm sides.

11.3 \, cm
GCSE Quiz True

16 \, cm
GCSE Quiz False

8 \, cm
GCSE Quiz False

12 \, cm
GCSE Quiz False

15 Pythagoras Theorem answer 6

 

\begin{aligned} a^2+b^2&=c^2\\ 8^2+8^2&=c^2\\ 64+64&=c^2\\ 128&=c^2\\ c&=\sqrt{128}\\ c&=11.3137085\\ c&=11.3 \, cm ~ \text{(3sf)} \end{aligned}

The diagonal of the square has a length of 11.3 centimeters.

7. ABC is an isosceles triangle.

 

15 Pythagoras Theorem question 7

Work out the height of the triangle.

8 \, cm
GCSE Quiz False

12 \, cm
GCSE Quiz True

8.31 \, cm
GCSE Quiz False

16.4 \, cm
GCSE Quiz False

15 Pythagoras Theorem answer 7

 

\begin{aligned} a^{2}&=c^{2}-b^{2}\\ a^{2}&=13^{2}-5^{2}\\ a^{2}&=169-25\\ a^{2}&=144\\ a&=\sqrt{144}\\ a&=12 \, cm \end{aligned}

8. ABCD is an isosceles trapezoid.

 

15 Pythagoras Theorem question 8

 

Work out the length of AD.

4.5 \, cm
GCSE Quiz False

12.5 \, cm
GCSE Quiz False

17 \, cm
GCSE Quiz True

23 \, cm
GCSE Quiz False

15 Pythagoras Theorem answer 8

 

\begin{aligned} a^{2}&=c^{2}-b^{2}\\ x^{2}&=7.5^{2}-6^{2}\\ x^{2}&=56.25-36\\ x^{2}&=20.25\\ x&=\sqrt{20.25}\\ x&=4.5 \, cm \end{aligned}

 

AD=4.5+8+4.5=17 \, cm

9. Here is a cm square grid. Calculate the distance between the points A and B.

 

15 Pythagoras Theorem question 9

9 \, cm
GCSE Quiz False

5.20 \, cm
GCSE Quiz False

3 \, cm
GCSE Quiz False

6.71 \, cm
GCSE Quiz True

15 Pythagoras Theorem answer 9

 

\begin{aligned} a^2+b^2&=c^2\\ 6^2+3^2&=c^2\\ 36+9&=c^2\\ 45&=c^2\\ c&=\sqrt{45}\\ c&=6.708203932\\ c&=6.7 \, cm ~ \text{(3sf)} \end{aligned}

10. Which is a right angled triangle?

 

15 Pythagoras Theorem question 10

A
GCSE Quiz False

B
GCSE Quiz True

C
GCSE Quiz False

D
GCSE Quiz False
\begin{aligned} \text{A: } 9^{2}+13^{2}&=250\\ 18^{2}&=324 \\ \end{aligned}

Not a right angled triangle because Pythagorean Theorem doesn’t work.

\begin{aligned} \text{B: } 8^{2}+15^{2}&=289\\ 17^{2}&=289 \\ \end{aligned}

Right angled triangle because Pythagorean Theorem works.

\begin{aligned} \text{C: } 9^{2}+19^{2}&=442\\ 23^{2}&=529 \\ \end{aligned}

Not a right angled triangle because Pythagorean Theorem doesn’t work.

\begin{aligned} \text{D: } 8^{2}+10^{2}&=164\\ 14^{2}&=196 \\ \end{aligned}

Not a right angled triangle because Pythagorean Theorem doesn’t work.

11. PQRS is made from two right angled triangles.

 

15 Pythagoras Theorem question 11

 

Work out the length of QR.

18.0 \, m
GCSE Quiz True

6 \, m
GCSE Quiz False

15.9 \, m
GCSE Quiz False

21.3 \, m
GCSE Quiz False

15 Pythagoras Theorem answer 11

 

Triangle \text{PQS:}

\begin{aligned} \\ b^{2}&=c^{2}-a^{2}\\ b^{2}&=10^{2}-8^{2}\\ b^{2}&=100-64\\ b^{2}&=36\\ b&=\sqrt{36}\\ b&=6 \, m \end{aligned}

Triangle \text{QRS}

\begin{aligned} \\ a^2+b^2&=c^2\\ 17^2+6^2&=c^2\\ 289+36&=c^2\\ 325&=c^2\\ c&=\sqrt{325}\\ c&=18.02775638\\ c&=18.0 \, m ~ \text{(3sf)} \end{aligned}

12. Here is a pattern made from right angled triangles. Work out the length x.

 

15 Pythagoras Theorem question 12

8.60 \, cm
GCSE Quiz False

16.6 \, cm
GCSE Quiz False

11.1 \, cm
GCSE Quiz True

8.66 \, cm
GCSE Quiz False

15 Pythagoras Theorem answer 12

 

Triangle \text{ABC:}

\begin{aligned} \\ a^2+b^2&=c^2\\ 5^2+7^2&=c^2\\ 74&=c^2\\ c&=\sqrt{74}\\ c&=8.602325267 \end{aligned}

 

Triangle \text{ACD:}

\begin{aligned} \\ a^2+b^2&=c^2\\ 5^2+8.60232567^2&=c^2\\ 99&=c^2\\ c&=\sqrt{99}\\ c&=9.949874371 \end{aligned}

13. Here is a pyramid.

 

15 Pythagoras Theorem question 13

 

Work out the height of the pyramid.

4.80 \, cm
GCSE Quiz False

16.3 \, cm
GCSE Quiz False

13.2 \, cm
GCSE Quiz False

10.7 \, cm
GCSE Quiz True

15 Pythagoras Theorem answer 13

 

\begin{aligned} a^{2}&=c^{2}-b^{2}\\ a^{2}&=12^{2}-5.5^{2}\\ a^{2}&=113.75\\ a&=\sqrt{113.75}\\ a&=10.6653645 \mathrm{cm} \\ a&=10.7 \, cm ~ \text{(3sf)} \end{aligned}

14. Here is a cuboid.

 

15 Pythagoras Theorem question 14

 

Work out the length AG.

Give your answer in its exact form.

\sqrt{58} \, cm
GCSE Quiz False

\sqrt{62} \, cm
GCSE Quiz True

\sqrt{53} \, cm
GCSE Quiz False

\sqrt{42} \, cm
GCSE Quiz False

15 Pythagoras Theorem answer 14 image 1

Length of \text{BG:}

\begin{aligned} \\ a^2+b^2&=c^2\\ 7^2+3^2&=c^2\\ 58&=c^2\\ c&=\sqrt{58}\\ c&=7.615773106 \end{aligned}

 

15 Pythagoras Theorem answer 14 image 2

 

Length of \text{AG:}

\begin{aligned} \\ a^2+b^2&=c^2\\ 2^2+7.615773106^2&=c^2\\ 62&=c^2\\ c&=\sqrt{62} \, cm \end{aligned}

15. Here is a right angled triangle.

Form an equation and use it to work out the value of x.

 

15 Pythagoras Theorem question 15

 

 

4 \, cm
GCSE Quiz False

10 \, cm
GCSE Quiz False

17 \, cm
GCSE Quiz False

12 \, cm
GCSE Quiz True
\begin{aligned} x^{2}+(x-7)^{2}&=(x+1)^{2}\\ x^{2}+x^{2}-7x-7x+49&=x^{2}+x+x+1\\ 2x^{2}-14x+49&=x^{2}+2x+1\\ x^{2}-16x-48&=0\\ (x-4)(x-12)&=0 \end{aligned}

 

x=4 \, or \, x=12

 

x cannot be 4 as you can’t have a negative side length so x=12

Pythagorean Theorem in 8th grade

In middle school, students…

  • prove the Pythagorean Theorem;
  • use the Pythagorean Theorem with trigonometric ratios to solve problems;
  • use the Pythagorean Theorem in proofs.

Pythagoras Theorem may feature in questions alongside other topics, such as trigonometry, circle theorems or algebra.

How do you do Pythagorean questions?

The Pythagorean Theorem is used to calculate a missing length in a right triangle . If you have a right angled triangle and you know two of the lengths, label the sides of the triangle a,b and c (c must be the hypotenuse – the longest side).

Pythagorean Theorem is a^2+b^2=c^2.

Substitute the values you know into Pythagorean Theorem and solve to find the missing side.

How do you find the hypotenuse of a question?

The hypotenuse of a right triangle is the longest side. If you know the lengths of the other two sides, you can find the length of the hypotenuse by squaring the two shorter sides, adding those values together and then taking the square root.

By doing this you are finding c in a^2+b^2=c^2

How do you find the missing side of a triangle?

If your triangle is a right triangle and you know two of the sides, you can use Pythagorean Theorem to find the length of the third side. To do this, label the sides a, b and c (with c being the hypotenuse – the longest side). Substitute the values you know into a^2+b^2=c^2 and solve to find the missing side.

Looking for more Pythagorean theorem math questions?

Try these:

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Why not learn more about how it works?

The content in this article was originally written by former UK Secondary teacher Beki Christian and has since been revised and adapted for US schools by elementary and middle school teacher Kathleen Epperson.

Beki Christian
Beki Christian
Maths Teacher and Author
Beki has 11 years experience teaching secondary Maths. She is currently working as a Maths tutor as well as writing resources and blog posts for Third Space Learning.
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15 Pythagoras Theorem Practice Problems

15 Pythagoras Theorem Practice Problems

Wish you could have the 15 Pythagoras Theorem questions from this blog in a ready-to-go worksheet? We've done just that!

DOWNLOAD FREE NOW

15 Pythagoras Theorem Practice Problems

Downloadable resource

Wish you could have the 15 Pythagoras Theorem questions from this blog in a ready-to-go worksheet? We've done just that!

DOWNLOAD FREE NOW
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