Pythagoras' theorem - Part 1 - KS3 Maths

Key points

A visual representation of Pythagoras’ Theorem. The image shows a right angled triangle. The right angled triangle has its right angled vertex in the bottom left corner. The vertical side to the left is labelled a, the horizontal side is labelled b, and the diagonal side is labelled c. Adjacent to each side of the triangle are three squares of different sizes. Each square shares a side with the triangle and has length the same size as the corresponding side of the triangle. Written left: the formula, a, squared plus b squared equals c squared. The right angled triangle is coloured green. The a, squared, and the square adjacent to a is coloured blue. The b squared, and the square adjacent to b is coloured orange. The c squared, and the square adjacent to c is coloured purple. — Image caption,
In Pythagoras' theorem 𝒂² + 𝒃² = 𝒄²

Pythagoras’ theorem states that for any right-angled triangle, the area of the square on the is equal to the sum of the areas of the squares on the other two sides.
It can be thought of as \(a\)² + \(b\)² = \(c\)² where \(a\) and \(b\) are the shorter sides of the triangle, and \(c\) is the hypotenuse (longest side).
Pythagoras’ theorem is only true for right-angled triangles. It is possible to check if a triangle is right-angled by in the lengths of the sides and seeing if the value of \(a\)² + \(b\)² is the same as the value of \(c\)².
Pythagoras’ theorem can be used to find a missing side of a right-angled triangle. To find the hypotenuse, the values of \(a\)² and \(b\)² into the equation, and solve to find \(c\). This will involve adding the two squares and finding the of the answer.
To find a shorter side, substitute the values into the equation and solve to find \(a\) or \(b\). This will involve subtracting the two squares and finding the square root of the answer.
An understanding of powers and roots is essential before exploring this topic.

What is Pythagoras' theorem?

Pythagoras’ theorem is a statement that is true for all right-angled triangles.It states that the area of the square on the is equal to the sum of the area of the squares on the other two sides.

It is useful to think of Pythagoras’ theorem as \(a\)² + \(b\)² = \(c\)².

The hypotenuse is labelled as \(c\) and the other two sides labelled as \(a\) and \(b\). This makes the areas of the squares \(c\)², \(a\)² and \(b\)².

Examples

Skip image gallery

Image caption,
The hypotenuse is the longest side of a right-angled triangle. It is always opposite the right angle. To label a right-angled triangle to use Pythagoras’ theorem, the hypotenuse should be labelled as 𝒄. The other two sides should be labelled 𝒂 and 𝒃. It does not matter which way round 𝒂 and 𝒃 are labelled.
Image caption,
The area of the blue square is 𝒂 x 𝒂, which is 𝒂². The area of the orange square is 𝒃 x 𝒃, which is 𝒃². The area of the purple square on the hypotenuse is 𝒄 x 𝒄, which is 𝒄². Pythagoras’ theorem states that for any right-angled triangle, the area of the square on the hypotenuse is equal to the sum of the area of the squares on the other two sides. Here the area of the blue square plus the area of the orange square equals the area of the purple square. In other words, 𝒂² + 𝒃² = 𝒄²
Image caption,
For this triangle, 𝒂 = 4, 𝒃 = 3, and 𝒄 = 5. The area of the blue square is 4 x 4, or 4². The area of the orange square is 3 x 3, or 3². The area of the purple square is 5 x 5, or 5²
Image caption,
Pythagoras’ theorem works for this triangle because the value of 𝒂² + 𝒃² is 25, and the value of 𝒄² is also 25. In other words, the area of the blue and the orange square added together is the same as the area of the purple square. Pythagoras’ theorem is only true for right-angled triangles. As 𝒂² + 𝒃² = 𝒄², the triangle must be a right-angled triangle.
Image caption,
This triangle looks like a right-angled triangle, but it might not be. To find out if it is a right-angled triangle check if Pythagoras’ theorem is true for the lengths of the sides.
Image caption,
To check if Pythagoras’ theorem is true for this triangle, the sides should be labelled 𝒂, 𝒃 and 𝒄. The longest side is always labelled 𝒄. It doesn’t matter which way round 𝒂 and 𝒃 are labelled. Here, 𝒂 is 2.1 cm, 𝒃 is 2 cm and 𝒄 is 2.9 cm.
Image caption,
From substituting in the lengths, the value of 𝒂² + 𝒃² is 2.1² + 2² = 8.41, and the value of 𝒄² is 2.9² which is also 8.41. 𝒂² + 𝒃² is the same as 𝒄². Pythagoras’ theorem is only true for right-angled triangles. As 𝒂² + 𝒃² = 𝒄², the triangle must be a right-angled triangle.
Image caption,
This triangle looks like a right-angled triangle, but it might not be. To find out if it is a right-angled triangle, check if Pythagoras’ theorem is true for the lengths of the sides.
Image caption,
Firstly, the sides 𝒂, 𝒃 and 𝒄 should be labelled. The longest side is always labelled 𝒄. It doesn’t matter which way round 𝒂 and 𝒃 are labelled. Here, 𝒂 is 9 mm, 𝒃 is 7 mm and 𝒄 is 11 mm.
Image caption,
Substitute in the lengths. The value of 𝒂² + 𝒃² is 9² + 7² = 130, and the value of 𝒄² is 11² which is 121. Pythagoras’ theorem is always true for right-angled triangles. 130 is not the same as 121. As 𝒂² + 𝒃² does not equal 𝒄², the triangle cannot be a right-angled triangle. (Please note that “≠” is the "not-equals" sign and the opposite of "=")

Question

Show that the triangle is a right-angled triangle using Pythagoras’ theorem.

An image of a right angled triangle. The triangle has sides of length, nine metres, twelve metres and fifteen metres. The triangle is coloured yellow.

Finding the length of the hypotenuse

can be used to calculate a missing side in a right-angled triangle. Follow these steps to find the length of the when the other two sides are given.

Label the sides \(a\), \(b\) and \(c\). Remember, the hypotenuse should always be labelled \(c\).
Then the values of \(a\) and \(b\) into the equation \(a\)² + \(b\)² = \(c\)².
Add the squares together to get the value of \(c\)².
Square root the value of \(c\)² to get the value of \(c\).

Examples

Skip image gallery

Image caption,
Find the length of LN
Image caption,
LN is the length from L to N. Label the sides 𝒂, 𝒃 and 𝒄. The hypotenuse is the longest side opposite the right angle, and should be labelled 𝒄. It does not matter which way round 𝒂 and 𝒃 are labelled.
Image caption,
Substitute the values into the equation 𝒂² + 𝒃² = 𝒄². 𝒂² = 5² and 𝒃² = 12². 𝒄² is not known and is the length LN that is needed, so keep this as 𝒄².
Image caption,
Calculate the value of the squares. 5² = 25, and 12² = 144. Add the squares together to get 169. This is the value of 𝒄².
Image caption,
The inverse of squaring is square rooting, so to find 𝒄, calculate the square root of 169. This gives the answer of 𝒄 = 13. Therefore LN = 13 cm.
Image caption,
Find the length PR to 1 decimal place.
Image caption,
PR is the length from P to R. Label the sides 𝒂, 𝒃 and 𝒄. The hypotenuse should be labelled 𝒄. It does not matter which way round 𝒂 and 𝒃 are labelled.
Image caption,
Substitute the values into the equation 𝒂² + 𝒃² = 𝒄². 𝒂² = 4.1² and 𝒃² = 9.2². 𝒄² is not known and is the length PR that is needed, so keep this as 𝒄².
Image caption,
Calculate the value of the squares with a calculator. 4.1² = 16.81 and 9.2² = 84.64. Add the squares together to get 101.45. This is the value of 𝒄².
Image caption,
The inverse (opposite) of squaring is square rooting, so to find 𝒄, calculate the square root of 101.45. This gives the answer of 𝒄 = 10.0722... Therefore, when rounded to 1 decimal place, PR = 10.1 cm.

Question

Find the length ST to 1 decimal place.

An image of a right angled triangle S, T, U. The right angle is at vertex U. Length S U is eleven metres. Length T U is three metres. Length S T has been highlighted orange. The triangle is coloured pink.

Finding the length of another side

Follow these steps to find the length of a side that is not the hypotenuse.

Label the sides \(a\), \(b\), and \(c\). Remember the should always be labelled \(c\)
Then the values of \(a\) and \(b\) into the equation \(a\)² + \(b\)² = \(c\)²
If finding \(b\), subtract the squares to get the value of \(b\)²
Finally, the value of \(b\)² to get the value of \(b\)

Examples

Skip image gallery

Image caption,
Find the length ST
Image caption,
ST is the length from S to T. Label the sides 𝒂, 𝒃 and 𝒄. The hypotenuse should be labelled 𝒄. It does not matter which way round 𝒂 and 𝒃 are labelled as the result will be the same. Substitute the values into the equation 𝒂² + 𝒃² = 𝒄². 𝒂² = 15² and 𝒄² = 25². 𝒃² is not known and is the length ST that is needed, so keep this as 𝒃². This leads to the equation 15² + 𝒃² = 25².
Image caption,
Calculate the value of the squares with a calculator. 15² = 225 and 25² = 625. This leads to the equation 225 + 𝒃² = 625
Image caption,
To work out the value of 𝒃², subtract 225 from both sides of the equation. This leads to the equation 𝒃² = 400
Image caption,
The inverse of squaring is square rooting, so to find 𝒃, calculate the square root of 400. This gives the answer of 𝒃 = 20. Therefore ST = 20 cm.
Image caption,
Find the length VW to 1 decimal place (d p).
Image caption,
VW is the length from V to W. Label the sides 𝒂, 𝒃 and 𝒄. The hypotenuse should be labelled 𝒄. It does not matter which way round 𝒂 and 𝒃 are labelled as the result will be the same. Substitute the values into the equation 𝒂² + 𝒃² = 𝒄². 𝒃² = 1.4² and 𝒄² = 3.6². 𝒂² is not known and is the length VW that is needed, so keep this as 𝒂². This leads to the equation 𝒂² + 1.4² = 3.6²
Image caption,
Calculate the value of the squares using a calculator. 1.4² = 1.96 , and 3.6² = 12.96. This leads to the equation 𝒂² + 1.96 = 12.96
Image caption,
To work out the value of 𝒂², subtract 1.96 from both sides of the equation. This leads to the equation 𝒂² = 11
Image caption,
The inverse of squaring is square rooting, so to find 𝒂, calculate the square root of 11. This gives the answer of 𝒂 = 3.3166… Therefore to 1 decimal place, VW = 3.3 m.

Question

Find the length EF.

An image of a right angled triangle D, E, F. The right angle is at vertex E. Length D E is ten metres. Length D F is twenty six metres. Length E F has been highlighted orange. The triangle is coloured green.

Practise using Pythagoras' theorem

Quiz

Practise calculating different lengths of sides using Pythagoras' theorem with this quiz. You may need a pen and paper to help you with your answers.

Real-life maths

An image of a television screen. The length of the diagonal, from one corner to the opposite corner, has been labelled as fifty five inches. — Image caption,
Pythagoras’ theorem can be used to calculate the diagonal size of the television screen.

Flat screen televisions are usually measured diagonally from opposite corners of the screen.

This means that a 55 inch television does not have a width of 55 inches.It would measure 55 inches from the top left corner to the bottom right corner (as opposite.)

Pythagoras’ theorem can be used to calculate the diagonal size of the television screen, if the width and the height are known.