Skip to content
Cambridge IGCSE0580

Surface area and volume

Mathematics 0580 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.

~7 min read

In this chapter

  1. 1Volume of a cuboid and a prism
  2. 2Surface area of a cuboid and a prism
  3. 3Volume and surface area of a cylinder
  4. 4Volume and surface area of a pyramid
  5. 5Volume and surface area of a cone
  6. 6Volume and surface area of a sphere
  7. 7Hemispheres and equal surface areas

1. Volume of a cuboid and a prism

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 cuboid the cross-section is a rectangle, so multiply the three edges. For a triangular prism, find the area of the triangle first.

Volume is measured in cubic units, such as cm³.

V_cuboid = l × w × h

V_prism = area of cross-section × length

Examiner insight

“The correct answer of 42 cm3 was rarely seen with the most common incorrect answer of 84 cm3 found by multiplying the length, width and height but not dividing by 2.”

Worked example 12 marks

A cuboid has length 10.8 cm, width 3.7 cm and height 4.6 cm. Work out the volume of the cuboid. Give your answer correct to 1 decimal place.

  1. 1

    V = 10.8 × 3.7 × 4.6 = 183.816 [1]

  2. 2

    V = 183.8 cm³ [1]

2. Surface area of a cuboid and a prism

Surface area is the total area of all the faces. It is measured in square units, such as cm².

A cuboid has three pairs of equal faces. Find each different face once, then double the sum.

For a prism, add the areas of every face, including the two ends.

A_cuboid = 2(lw + lh + wh)

Examiner insight

“A very common error was thinking there were four of one face area and two of another.” “Some did not double at the relevant stage.”

Worked example 13 marks

A cuboid has length 10 cm, width 3 cm and height 4 cm. Work out the surface area of the cuboid.

  1. 1

    Faces: 10 × 3 = 30, 10 × 4 = 40, 3 × 4 = 12 [1]

  2. 2

    Sum of the three faces = 82, then double: 2 × 82 [1]

  3. 3

    A = 164 cm² [1]

3. Volume and surface area of a cylinder

A cylinder is a prism with a circular cross-section, so its volume is the area of the circle times the height.

Its curved surface area is 2π r h. Its total surface area adds the two circular ends.

Answers may be asked for in terms of π. Leave π in the answer and do not use a decimal.

V = π r² h

A_curved = 2π r h

A_total = 2π r h + 2π r² = 2π r(r + h)

Examiner insight

“Most just took the curved surface area or the area of one or two ends as the total surface area.” “Whilst many candidates were able to correctly use the formula for the volume of the cylinder, only a minority of these went on to give their answer in terms of π, with the majority giving a decimal answer.”

Worked example 13 marks

Do not use a calculator. A cylinder has radius 3 cm and height 10 cm. Find its total surface area. Give your answer in terms of π.

  1. 1

    Curved surface: 2π × 3 × 10 = 60π [1]

  2. 2

    Two ends: 2 × π × 3² = 18π [1]

  3. 3

    A = 78π cm² [1]

4. Volume and surface area of a pyramid

A pyramid has a base and triangular faces that meet at one point. Its volume is one third of the base area times the perpendicular height.

The perpendicular height goes straight up from the base to the point. It is not the sloping edge.

For surface area, add the base and every triangular face.

V = 1/3 × base area × perpendicular height

Examiner insight

“Common errors included the assumption that either the slant edge of the triangle or the height of the pyramid were the height of the triangle and occasionally that the triangular face was equilateral.”

Worked example 12 marks

ABCDE is a square-based pyramid. The base BCDE has side BC = 16.5 cm. The perpendicular height of the pyramid is AF = 28 cm. Calculate the volume of the pyramid.

  1. 1

    V = 1/3 × 16.5² × 28 [1]

  2. 2

    V = 2541 cm³ [1]

5. Volume and surface area of a cone

A cone has a vertical height h and a slant height l, the length of the sloping edge. The curved surface area uses the slant height.

The volume is one third of π r² h, using the vertical height.

The total surface area adds the circular base.

V = 1/3π r² h

A_curved = π r l

A_total = π r l + π r²

Examiner insight

“The most common error was to find the area of the curved surface and omit the area of the base.” “A few did not use the slant height for the area of the curved surface but calculated the vertical height of the cone and used this in the formula.”

Worked example 12 marks

A cone has a radius of 7 cm and a height of 8 cm. Work out the volume of the cone.

  1. 1

    V = 1/3 × π × 7² × 8 [1]

  2. 2

    V = 410.5... = 411 cm³ [1]

6. Volume and surface area of a sphere

For a sphere the volume has r³ and the surface area has r². Check which one the question asks for.

To find r from a volume, divide, then take the cube root. To change cm³ into litres, divide by 1000.

When the volume is given in terms of π, cancel π first. The working then stays exact.

V = 4/3π r³

A = 4π r²

Examiner insight

“It was evident that some candidates were using square root for their final step rather than the needed cube root.” “Few candidates showed the explicit step to convert cm3 to litres of division by 1000.”

Worked example 12 marks

Do not use a calculator. A sphere has volume 36π cm³. Find the radius of the sphere.

  1. 1

    4/3π r³ = 36π, so r³ = 36 × 3/4 = 27 [1]

  2. 2

    r = 3 cm [1]

7. Hemispheres and equal surface areas

A hemisphere is half a sphere. Its curved surface is half of 4π r².

A total surface area also includes the flat circular face, π r². So the total is 3π r².

To compare two solids, write each surface area, then equate them. Each solid keeps its own radius.

A_hemisphere, curved = (4π r²)/2 = 2π r²

A_hemisphere, total = 2π r² + π r² = 3π r²

Examiner insight

“A common error was not halving the hemisphere.” “Some candidates missed out the circle bases altogether for each shape.”

Worked example 13 marks

The cone has radius r cm and sloping edge 12 cm. The hemisphere has radius 4 cm. The total surface area of the cone is equal to the total surface area of the hemisphere. Show that r² + 12r - 48 = 0.

  1. 1

    Cone: π r² + π r × 12 [1]

  2. 2

    Hemisphere: (4π × 4²)/2 + π × 4² = 32π + 16π = 48π [1]

  3. 3

    π r² + 12π r = 48π. Cancel π: r² + 12r - 48 = 0 [1]

Recap

  • Prism volume = area of cross-section × length. Surface area = sum of all faces.
  • Cuboid surface area: three pairs of faces, so double the sum.
  • Cylinder: V = π r² h. Total area = 2π r h + 2π r².
  • Pyramid: V = 1/3 × base area × perpendicular height.
  • Cone: V = 1/3π r² h. Curved area = π r l with the slant height l.
  • Sphere: V = 4/3π r³, A = 4π r². A hemisphere's total area is 3π r².
  • Extended: the same solids in longer problems, such as equating two surface areas.

End-of-chapter exercise

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

  1. A cuboid measures 7.5 cm by 4.2 cm by 3.0 cm. Calculate its volume.2 marks
  2. Do not use a calculator. A cuboid is 6 cm long, 4 cm wide and 2 cm high. Work out its total surface area.3 marks
  3. Do not use a calculator. A prism has a right-angled triangle as its cross-section. The shorter sides of the triangle are 6 cm and 8 cm. The prism is 10 cm long. Work out its volume.3 marks
  4. A cylinder has diameter 10 cm and height 12 cm. Calculate its volume. Give your answer correct to 3 significant figures.3 marks
  5. Do not use a calculator. A pyramid has a square base of side 6 cm. Its perpendicular height is 5 cm. Work out its volume.3 marks
  6. Do not use a calculator. A cylinder has radius 3 cm and height 4 cm. Find its total surface area. Give your answer in terms of π.3 marks
  7. A cone has base radius 6 cm, vertical height 8 cm and sloping edge 10 cm. Calculate its total surface area. Give your answer correct to 3 significant figures.3 marks
  8. A spherical tank has radius 50 cm. Calculate its capacity in litres. Give your answer correct to 3 significant figures.3 marks
  9. Do not use a calculator. A solid metal cylinder has radius 6 cm and height 8 cm. It is melted down and all the metal is made into one solid sphere. Find the decrease in the total surface area. Give your answer in terms of π.5 marks
  10. Do not use a calculator. A solid cylinder has radius r cm and height 5 cm. A solid hemisphere has radius 4 cm. Each solid is painted all over, and the two painted areas are equal. Find r, then find the volume of the cylinder in terms of π.6 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