Geometry

To calculate the length of a side on a right-angled triangle when you know the sizes of the other two, you need to use Pythagoras' Theorem.

Pythagoras' Theorem says that, in a right angled triangle:

The square of the hypotenuse is equal to the sum of the squares on the other two sides.

Pythagoras diagram showing right angled triangle with values a, b and c and squares a≤, b≤ and c≤

We can write this more simply as :

{a^2} = {b^2} + {c^2}

Diagram of right angled triangle with a, b and c dimensions and the formula a² = b² + c²

Calculating the length of the hypotenuse

Question

Use Pythagoras' Theorem to calculate the length of the hypotenuse. Give your answer to 2 decimal places.

rectangle 4 x 7 x X

Write the equation x^{2} = 7^{2} + 4^{2}

Square the lengths you know x^{2} = 49 + 16

Add together x^{2} = 65

Find the square root x = \sqrt {65}

x = 8.06 (to\,2\,d.p.)

Question

Calculate the length of side x

(Give your answer to 2 decimal places)

Diagram of a 5 x 9 right-angled triangle with the gradient edge marked x

x^{2} = 5^{2} + 9^{2}

x^{2} = 25 + 81

x^{2} = 106

x = \sqrt {106}

x = 10.30 (to\, 2\,d.p.)

Example

Calculate the length of the side marked a.

Give your answer to 2 decimal places.

rectangle 12 x 8 x a

Answer

  • Write the equation: {12^2} = {a^2} + {8^2}
  • Organise the equation {a^2} = {12^2} - {8^2}. To find the length of a short side, we can also use the formula {b^2} = {a^2} - {c^2}
  • Square the lengths you know: {a^2} = 144 - 64
  • Do the subtraction: {a^2} = 80
  • Find the square root: a = \sqrt {80}
  • a = 8.94\,(to\,2\,d.p.)

Question

Calculate the length of side a.

Give your answer to 2 decimal places

Right-angled triangle with values a, 9 and 13

a^{2} = 13^{2} - 9^{2}

a^{2} = 169 - 81

a^{2} = 88

a = \sqrt 88

a = 9.38