Learning Objectives
  • Convert between metric units of length, area, volume and capacity.
  • Calculate volumes of cuboids, prisms and cylinders.
  • Calculate total and curved surface areas of prisms and cylinders.
  • Use nets and cross-sections to organise surface-area calculations.
  • Solve reverse and compound mensuration problems involving prisms and cylinders.
Key Terms
Prism
A solid with a uniform cross-section throughout its length.
Cross-section
The repeated face obtained by slicing perpendicular to the length of a prism.
Cylinder
A circular prism.
Curved surface area
The area of the curved outer surface of a cylinder.
Total surface area
The sum of all exposed surfaces.
Capacity
The amount a container can hold, commonly measured in millilitres or litres.
8th Edition Chapter Map
  • Units and conversions
  • Volumes of cuboids and prisms
  • Volume of a cylinder
  • Surface area of prisms
  • Curved and total surface area of cylinders
Units Of Length, Area And Volume

When a length conversion factor is k, the area conversion factor is k^2 and the volume conversion factor is k^3. Since 1\text{ m}=100\text{ cm}, 1\text{ m}^2=10000\text{ cm}^2 and 1\text{ m}^3=1000000\text{ cm}^3.

Conversion Result
1\text{ cm}^3 1 ml
1000\text{ cm}^3 1 litre
1\text{ m}^3 1000 litres
1\text{ m}^2 10 000 cm^2
Volume Of A Prism

Every prism has a constant cross-sectional area. Multiply that area by the perpendicular length of the prism.

V=\text{area of cross-section}\times\text{length}
Worked Example: Triangular Prism

Question: A triangular prism has a right-triangular cross-section with perpendicular sides 6 cm and 8 cm. Its length is 15 cm. Find the volume.

  1. Cross-sectional area =\frac12(6)(8)=24 cm^2.
  2. Volume =24\times15=360 cm^3.

Answer: 360 cm^3.

Volume Of A Cylinder

A cylinder is a prism with circular cross-section. Use radius, not diameter.

V=\pi r^2h
Worked Example: Cylinder Volume

Question: Find the volume of a cylinder of diameter 10 cm and height 18 cm.

  1. Radius =5 cm.
  2. V=\pi(5)^2(18)=450\pi cm^3.
  3. 450\pi\approx1413.72.

Answer: 450\pi cm^3, approximately 1414 cm^3.

Surface Area Of A Prism

Use a net or organise faces systematically. For a right prism, the lateral area equals perimeter of cross-section multiplied by length. Add the two congruent end faces.

\text{surface area}=2(\text{cross-sectional area})+(\text{perimeter of cross-section})(\text{length})
Worked Example: Triangular Prism Surface Area

Question: A triangular prism has cross-section sides 5 cm, 5 cm and 6 cm, cross-sectional area 12 cm^2, and length 10 cm. Find its total surface area.

  1. Two triangular ends: 2(12)=24 cm^2.
  2. Lateral area: (5+5+6)(10)=160 cm^2.
  3. Total =184 cm^2.

Answer: 184 cm^2.

Curved Surface Area Of A Cylinder

Unrolling the curved surface gives a rectangle with length equal to the circumference 2\pi r and width equal to the cylinder height h.

\text{curved surface area}=2\pi rh
\text{total surface area}=2\pi rh+2\pi r^2
Worked Example: Closed Cylinder

Question: Find the total surface area of a closed cylinder with radius 4 cm and height 11 cm.

  1. Curved area =2\pi(4)(11)=88\pi.
  2. Two circular ends =2\pi(4^2)=32\pi.
  3. Total =120\pi cm^2.

Answer: 120\pi cm^2, approximately 377 cm^2.

Open Containers

Read whether the solid is open at one end, both ends or closed. An open cylinder may require only one circular base; a pipe may require neither end but may have inner and outer curved surfaces.

Reverse Problems
Worked Example: Find A Missing Height

Question: A cylinder has volume 972\pi cm^3 and radius 9 cm. Find its height.

  1. 972\pi=\pi(9^2)h.
  2. 972=81h.
  3. h=12.

Answer: 12 cm.

Compound Solids And Removed Parts

For a compound solid, add volumes of joined parts or subtract removed parts. For surface area, count only exposed faces; surfaces where solids are joined are internal and should not be included.

Examination Guidance
  • Convert all dimensions to the same unit before calculating.
  • Draw or imagine a net for surface-area questions.
  • Use radius, not diameter, in circle formulas.
  • Keep answers in terms of \pi when requested.
Common Mistakes
  • Using the linear conversion factor for area or volume.
  • Including internal joined faces in a compound solid’s external surface area.
  • Forgetting one or both circular ends of a cylinder.
  • Using sloping length instead of perpendicular prism length.
Knowledge Check And Practice

1. Find the volume of a cuboid 8 cm by 5 cm by 12 cm.

Answer: 480 cm^3.

2. A prism has cross-sectional area 35 cm^2 and length 14 cm. Find its volume.

Answer: 490 cm^3.

3. Find the volume of a cylinder with radius 3 cm and height 10 cm.

Answer: 90\pi cm^3.

4. Find the curved surface area of a cylinder with radius 5 cm and height 7 cm.

Answer: 70\pi cm^2.

5. Convert 2.4 litres to cm^3.

Answer: 2400 cm^3.
Extended Worked Practice
Finding Cylinder Height From Surface Area

Question: A closed cylinder has radius 4 cm and total surface area 176\pi cm squared. Find its height.

  1. 2\pi r^2+2\pi rh=176\pi.
  2. 32\pi+8\pi h=176\pi.
  3. 8h=144, so h=18.

Answer: 18 cm.

Volume Of A Trapezium Prism

Question: The cross-section of a prism is a trapezium with parallel sides 5 cm and 11 cm and perpendicular height 6 cm. The prism length is 20 cm. Find its volume.

  1. Cross-sectional area =\frac12(5+11)(6)=48 cm squared.
  2. Volume =48(20)=960 cm cubed.

Answer: 960 cm cubed.

Capacity Of A Cylindrical Tank

Question: A cylindrical tank has radius 0.8 m and length 2.5 m. Find its capacity in litres.

  1. V=\pi(0.8)^2(2.5)=1.6\pi\approx5.0265 cubic metres.
  2. 1 cubic metre =1000 litres.

Answer: Approximately 5030 litres to 3 significant figures.

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