Skip to content
Cambridge IGCSE0607

Surface area and volume

International Mathematics 0607 Chapter Notes

What this chapter covers

Surface area and volume
ShareWhatsAppPost
Surface area and volume notes

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

Read the notes

The full Surface area and volume notes as text: skim, search, and jump between subtopics.

~8 min read

In this chapter

  1. 1Cuboids and cubes
  2. 2Prisms
  3. 3Cylinders
  4. 4Cones and the slant height
  5. 5Spheres
  6. 6Pyramids
  7. 7Solids that are combined or equal in volume

1. Cuboids and cubes

Volume is measured in cm³. Surface area is the total area of all the faces, in cm².

A cuboid has six rectangular faces in three equal pairs. A cube has six identical square faces, so its total surface area is six times the area of one face.

To go back from a surface area to a side, divide by 6, then take the square root.

V = l × w × h

surface area of cuboid = 2(lw + lh + wh)

surface area of cube = 6l², V = l³

Examiner insight

“Some candidates divided by 8 instead of 6.” “Some candidates tried to use the answer and work backwards.”

Worked example 13 marks

The total surface area of the cube is 384 cm². Show that the volume of the cube is 512 cm³.

  1. 1

    one face = 384 ÷ 6 = 64 cm² [1]

  2. 2

    side = √64 = 8 cm [1]

  3. 3

    volume = 8³ = 512 cm³ [1]

2. Prisms

A prism has the same cross-section all along its length. Its volume is the area of the cross-section multiplied by the length.

For a triangular prism the cross-section is a triangle, so remember the half in its area.

For surface area, add up every face: the two ends and the rectangles along the length. Check that you have counted each face once.

V = area of cross-section × length

triangle area = 1/2 × base × height

Examiner insight

“This was not a particularly well answered part with many candidates forgetting to take into account that it was a triangular prism and omitting to divide their volume by 2 leaving a common final incorrect final answer of 264.”

Worked example 15 marks

Do not use a calculator. A prism has a right-angled triangle as its cross-section. The shorter sides of the triangle are 3 cm and 4 cm and the longest side is 5 cm. The prism is 10 cm long. Find the volume and the total surface area of the prism.

  1. 1

    cross-section = 1/2 × 3 × 4 = 6 cm² [1]

  2. 2

    volume = 6 × 10 = 60 cm³ [1]

  3. 3

    two ends: 2 × 6 = 12 cm² [1]

  4. 4

    three rectangles: (3 + 4 + 5) × 10 = 120 cm² [1]

  5. 5

    surface area = 12 + 120 = 132 cm² [1]

3. Cylinders

The formulas for the curved surface area and the volume of a cylinder are given on the paper.

A cylinder with two circular ends has a total surface area of the curved surface plus two circles.

Answers may be asked for in terms of π. When π appears on both sides of an equation, it cancels, so there is no need for a decimal value.

curved surface area = 2π r h

V = π r² h

total surface area = 2π r h + 2π r²

Examiner insight

“Some worked with a decimal value for π which is unnecessary as π cancels on both sides of the equation.” “Some used 48 or 48π as the radius in the formula to find the curved surface area.”

Worked example 12 marks

A cylinder has height 6 cm. The curved surface area of the cylinder is 48π cm². Find the radius of the cylinder.

  1. 1

    2 × π × r × 6 = 48π, so 12π r = 48π [1]

  2. 2

    r = 48 ÷ 12 = 4 cm [1]

4. Cones and the slant height

The formulas for the curved surface area and the volume of a cone are given on the paper.

The curved surface area uses the slant height l, the sloping edge, and not the vertical height h. Find l with Pythagoras.

curved surface area = π r l

V = 1/3 π r² h

l² = h² + r²

Examiner insight

“Some candidates used the vertical height of the cone instead of its slant height when finding the curved surface area of the cone.” “Many simply rewrote the formula for the curved surface area of a cone as πrh and were not able to gain any marks.”

Worked example 12 marks

A cone has a volume of 48π cm³. The radius of the cone is 6 cm. Find the height of the cone.

  1. 1

    1/3 × π × 6² × h = 48π, so 12π h = 48π [1]

  2. 2

    h = 48 ÷ 12 = 4 cm [1]

5. Spheres

The formulas for the surface area and the volume of a sphere are given on the paper.

The radius is half of the diameter. In the volume the radius is cubed, and in the surface area it is squared.

On a graphic display calculator, enter the whole expression in one go and read off the result. Round only at the end.

surface area = 4π r²

V = 4/3 π r³

Examiner insight

“Errors were seen when candidates forgot to halve the sphere and mixed up the use of the radius squared or cubed.” “A common error was to find the volume of the sphere but forget to halve this value.”

Worked example 12 marks

A sphere has radius 7.3 cm. Work out the surface area of this sphere.

  1. 1

    surface area = 4 × π × 7.3² [1]

  2. 2

    = 669.66… ≈ 670 cm² [1]

6. Pyramids

The formula for the volume of a pyramid is given on the paper.

Work out the area of the base first. For a square base of side s it is s². Then multiply by the height and by one third.

V = 1/3 × base area × height

Examiner insight

“This was not a simple case of substitution into the correct formula as the area of the base had to be worked out from the information in the question.” “Many ignored this stage so their answer was 70 cm3.”

Worked example 13 marks

Do not use a calculator. A pyramid has height h cm. The base of the pyramid is a square with side 10 cm. The volume of the pyramid is 900 cm³. Work out the value of h.

  1. 1

    base area = 10 × 10 = 100 cm² [1]

  2. 2

    1/3 × 100 × h = 900 [1]

  3. 3

    h = 900 × 3 ÷ 100 = 27 [1]

7. Solids that are combined or equal in volume

For a compound solid, list the surfaces that are on the outside, then add them. A face that is joined to another solid is not part of the surface.

When two solids have equal volumes, write each volume with its formula and put them equal. Cancel any common factors, including π, before you rearrange.

Keep accurate values until the final answer.

Examiner insight

“Many candidates used a formula for the total surface area of the cylinder, not dealing correctly with the area of the top surface of the cylinder.” “Premature rounding or forgetting to add the area of the base often resulted in the final answer being inaccurate or incorrect.”

Worked example 15 marks

In this question, all lengths are in centimetres. A cone has radius 5x and sloping edge 13x. A sphere has radius r. The volume of the cone is equal to the volume of the sphere. Show that r³ = 75x³.

  1. 1

    height of cone: h² = (13x)² - (5x)² [1]

  2. 2

    h = 12x [1]

  3. 3

    1/3 π (5x)² (12x) = 4/3 π r³ [2]

  4. 4

    100π x³ = 4/3 π r³, so r³ = 75x³ [1]

Recap

  • Volume is in cm³. Surface area is in cm². Do not mix them up.
  • Cube: surface area 6l². To find the side from a surface area, divide by 6 first.
  • Prism: volume = cross-section × length. A triangle has a half.
  • Cylinder: curved surface area 2π r h, volume π r² h. Add 2π r² for the two ends.
  • Cone: curved surface area π r l uses the slant height. Find l with l² = h² + r².
  • Sphere: surface area 4π r², volume 4/3 π r³. Halve the volume for a hemisphere.
  • Pyramid: volume 1/3 × base area × height.
  • Leave π in the answer when asked. Round only at the end.

End-of-chapter exercise

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

  1. Do not use a calculator. A cuboid measures 5 cm by 4 cm by 3 cm. Find its volume.1 mark
  2. A cuboid measures 5 cm by 4 cm by 3 cm. Find its total surface area.2 marks
  3. Do not use a calculator. A prism is 10 cm long. Its cross-section is a triangle with base 6 cm and perpendicular height 8 cm. Find the volume of the prism.2 marks
  4. Do not use a calculator. A cylinder has radius 5 cm and height 8 cm. Find its curved surface area. Give your answer in terms of π.2 marks
  5. A cylinder has radius 4.5 cm and height 12 cm. It has two circular ends. Calculate its total surface area. Give your answer correct to 3 significant figures.3 marks
  6. A cone has base radius 3.5 cm and vertical height 12 cm. Calculate its curved surface area. Give your answer correct to 3 significant figures.3 marks
  7. Do not use a calculator. A solid sphere has diameter 6 cm. Find its volume. Give your answer in terms of π.2 marks
  8. Do not use a calculator. A pyramid has a square base of side 6 cm. Its volume is 120 cm³. Find the height of the pyramid.3 marks
  9. Do not use a calculator. A jeweller melts a solid silver cylinder of radius 4 cm and height 6 cm. All of the silver is used to make one solid cone of base radius 6 cm. Find, in terms of π, how much greater the total surface area of the cone is than the total surface area of the cylinder.6 marks
  10. A candle holder is a solid metal cylinder of radius 5 cm and height h cm. A hole in the shape of a cone is cut into the centre of its top face. The cone has radius 3 cm and vertical height 4 cm. The total surface area of the holder, including the inside of the hole, is 500 cm². Calculate the volume of metal in the holder. Give your answer correct to 3 significant figures.7 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