Skip to content
Cambridge IGCSE0580

Pythagoras’ theorem

Mathematics 0580 Chapter Notes

What this chapter covers

Pythagoras’ theorem
ShareWhatsAppPost
Pythagoras’ theorem notes

Read online free. Download a watermarked copy with a free account.

Read the notes

The full Pythagoras’ theorem notes as text: skim, search, and jump between subtopics.

~5 min read

In this chapter

  1. 1The theorem and the hypotenuse
  2. 2Finding the hypotenuse
  3. 3Finding a shorter side
  4. 4Right-angled triangles inside other shapes
  5. 5Accuracy and checking

1. The theorem and the hypotenuse

Pythagoras' theorem applies to a right-angled triangle only. The hypotenuse is the side opposite the right angle. It is always the longest side.

The square on the hypotenuse equals the sum of the squares on the two shorter sides.

Check that the triangle has a right angle before you use the theorem.

c² = a² + b²

Examiner insight

“A significant number attempted to apply Pythagoras’ theorem but candidates should have realised that with no right angle indicated, even if one angle looked close to 90°, this was not going to be appropriate.” “Another error was to assume the triangles had a right angle and so Pythagoras’ theorem was applied.”

Worked example 13 marks

Do not use a calculator. A right-angled triangle has shorter sides 6 cm and 8 cm. Work out the length of the hypotenuse.

  1. 1

    hypotenuse² = 6² + 8² [1]

  2. 2

    = 36 + 64 = 100 [1]

  3. 3

    hypotenuse = √(100) = 10 cm [1]

2. Finding the hypotenuse

To find the hypotenuse, square both shorter sides and add them. Then take the square root.

The square root is the last step. Do not stop at the sum of the squares.

c = √(a² + b²)

Worked example 12 marks

A right-angled triangle has shorter sides 11 cm and 18 cm. Calculate the hypotenuse, x.

  1. 1

    x² = 11² + 18² = 445 [1]

  2. 2

    x = √(445) = 21.09… = 21.1 cm [1]

3. Finding a shorter side

To find a shorter side, subtract the square of the known shorter side from the square of the hypotenuse. Then take the square root.

A shorter side is always smaller than the hypotenuse.

a = √(c² - b²)

Examiner insight

“Many candidates were able to correctly apply Pythagoras’ theorem with only a small number adding rather than subtracting.” “There was no indication that answers were checked to see whether they were reasonable, so answers that were longer than the hypotenuse were offered with no attempt to check or correct these.”

Worked example 13 marks

The point A is on the top of a vertical building. The horizontal distance from the base of the building to B is 22 m. AB = 25 m and the height of the building is h m. Find the value of h.

  1. 1

    AB is the hypotenuse: h² = 25² - 22² [1]

  2. 2

    h² = 625 - 484 = 141 [1]

  3. 3

    h = √(141) = 11.87… = 11.9 [1]

4. Right-angled triangles inside other shapes

Look for a right angle in a rectangle, a trapezium or an isosceles triangle. Draw the right-angled triangle you need.

In an isosceles triangle, the height meets the base at its midpoint. Halve the base first.

A rectangle's diagonal is the hypotenuse of a triangle made from two of its sides.

Examiner insight

“Although candidates recognised the need to use Pythagoras’ theorem, several did not halve the base and used the value 4 rather than 2.”

Worked example 14 marks

Do not use a calculator. An isosceles triangle has equal sides of 13 cm and a base of 10 cm. Work out the perpendicular height of the triangle.

  1. 1

    Half the base = 10 ÷ 2 = 5 cm [1]

  2. 2

    h² = 13² - 5² [1]

  3. 3

    = 169 - 25 = 144 [1]

  4. 4

    h = √(144) = 12 cm [1]

5. Accuracy and checking

Write the unrounded value before you round. Give 3 significant figures unless the question says otherwise.

Then check. A hypotenuse must be longer than both shorter sides. A shorter side must be shorter than the hypotenuse.

Examiner insight

“Some candidates only gained partial credit as an answer of 8.9 was given without a more accurate answer shown.” “A common error was to give the answer 4.6 without showing a more accurate value; answers should be given to 3 significant figures.”

Worked example 13 marks

A right-angled triangle has shorter sides 5 cm and 8 cm. Calculate the hypotenuse. Give your answer correct to 3 significant figures.

  1. 1

    c² = 5² + 8² = 89 [1]

  2. 2

    c = √89 = 9.4339… [1]

  3. 3

    c = 9.43 cm (longer than 8 cm, so reasonable) [1]

Recap

  • Pythagoras' theorem applies to right-angled triangles only.
  • The hypotenuse is opposite the right angle and is the longest side.
  • Hypotenuse: add the squares, then take the square root.
  • Shorter side: subtract the squares, then take the square root.
  • In an isosceles triangle, halve the base to make a right-angled triangle.
  • Show the unrounded value, then give 3 significant figures, and check the size.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. State which side of a right-angled triangle is the hypotenuse.1 mark
  2. Do not use a calculator. A right-angled triangle has shorter sides 9 cm and 12 cm. Find the hypotenuse.2 marks
  3. Do not use a calculator. A right-angled triangle has hypotenuse 17 cm and one shorter side 8 cm. Find the other shorter side.3 marks
  4. A right-angled triangle has shorter sides 6.4 cm and 8.7 cm. Calculate the hypotenuse. Give your answer correct to 3 significant figures.3 marks
  5. A ladder 4.2 m long leans against a vertical wall. Its foot is on horizontal ground 1.5 m from the wall. Calculate how high up the wall the ladder reaches. Give your answer correct to 3 significant figures.3 marks
  6. Do not use a calculator. A rectangle is 24 cm long and 7 cm wide. Work out the length of a diagonal.3 marks
  7. Do not use a calculator. An isosceles triangle has equal sides of 26 cm and a base of 20 cm. Work out its perpendicular height.4 marks
  8. A right-angled triangle has hypotenuse 13.6 cm and one shorter side 8.2 cm. Calculate the area of the triangle. Give your answer correct to 3 significant figures.4 marks
  9. Do not use a calculator. A window pane is a rhombus. The strip around its four edges has a total length of 52 cm. One diagonal of the pane is 24 cm long. The diagonals of a rhombus cross at right angles and cut each other in half. Work out the length of the other diagonal.4 marks
  10. The four straight sides of a field are AB, BC, CD and DA. Angle ABC = 90° and angle ADC = 90°. AB = 60 m, BC = 75 m and CD = 42 m. Calculate the length of AD. Give your answer correct to 3 significant figures.5 marks

Free notes are just the start.

Unlock every Workbook and Chapter at a Glance, and generate your own worksheets and predicted papers.

Explore plans

Related chapters

Types of numberSetsPowers and rootsFractions, decimals and percentagesOrderingThe four operations