Study Companion
Topic 08 | Measurement & Space

Volume & capacity

Year 7 core: volumes of right prisms (rectangular and triangular), capacity units, and the link between volume and capacity.

40-50 min Printable practice Answer key
VC2M7M02
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Year 7 core

By the end of this topic you should be able to:

1. Volume and capacity

Volume is the amount of three-dimensional space a solid occupies. Units are cubed: mm^3, cm^3, m^3.

Capacity is how much liquid a container holds. Units are mL, L, kL.

The two are linked:   1\;1 cm^3 of water is exactly 11 mL.

Conversions

Volume units
1 cm3=1000 mm3,1 m3=1 000 000 cm3.1 \text{ cm}^{3} = 1000 \text{ mm}^{3}, \quad 1 \text{ m}^{3} = 1\,000\,000 \text{ cm}^{3}.
Capacity units
1 L=1000 mL,1 kL=1000 L.1 \text{ L} = 1000 \text{ mL}, \quad 1 \text{ kL} = 1000 \text{ L}.
Linking volume and capacity
1 cm3=1 mL,1 m3=1000 L=1 kL.1 \text{ cm}^{3} = 1 \text{ mL}, \qquad 1 \text{ m}^{3} = 1000 \text{ L} = 1 \text{ kL}.

2. Volume of a right prism

A right prism has two identical, parallel end-faces (the bases) joined by rectangular side-faces. For any right prism, the volume is the area of its base multiplied by its length.

Volume of any right prism
V  =  area of base×length.V \;=\; \text{area of base} \times \text{length}.

3. Rectangular prism (cuboid)

A rectangular prism has a rectangular base.

LHW
A rectangular prism (cuboid) has length L, width W and height H.
Rectangular prism
V=L×W×H.V = L \times W \times H.
Worked example 1 Volume of a box

A box measures 88 cm long, 55 cm wide, 44 cm high. Find its volume.

V=8×5×4=160 cm3.V = 8 \times 5 \times 4 = 160 \text{ cm}^{3}.

How many millilitres of water could this box hold?

160 cm3=160 mL.160 \text{ cm}^{3} = 160 \text{ mL}.

4. Triangular prism

A triangular prism has a triangular base. Calculate the area of the triangle, then multiply by the length of the prism.

Triangular prism
V=12×b×h×ℓ,V = \tfrac{1}{2} \times b \times h \times \ell,

where bb and hh are the base and perpendicular height of the triangle, and ℓ\ell is the length of the prism.

Worked example 2 Volume of a tent

A ridge tent has a triangular cross-section of base 2.42.4 m and height 1.81.8 m, and is 33 m long. Find the volume.

  1. Area of triangle: 12×2.4×1.8=2.16\tfrac{1}{2} \times 2.4 \times 1.8 = 2.16 m^2.
  2. Volume: 2.16×3=6.482.16 \times 3 = 6.48 m^3.

5. Finding a missing dimension

Worked example 3 Finding height from a capacity

A rectangular tank has a base 22 m by 1.51.5 m and holds 90009000 L of water. What is its depth?

  1. Convert capacity: 90009000 L =9= 9 m^3.
  2. V=L×W×HV = L \times W \times H, so 9=2×1.5×H9 = 2 \times 1.5 \times H.
  3. 9=3H9 = 3H, so H=3H = 3 m.

Practice: Year 7 core

Fluency

Unit conversions

    1. Convert 25002500 mL to litres.
    2. Convert 4.64.6 L to mL.
    3. Convert 75007500 cm^3 to mL.
    4. Convert 33 m^3 to litres.
    5. Convert 45004500 mm^3 to cm^3.
    6. A 22 L bottle holds how many cm^3?
Fluency

Rectangular prism

    1. Find the volume of a 6×5×46 \times 5 \times 4 cm cuboid.
    2. Find the volume of a cube with edge 77 cm.
    3. Find the volume of a 12×8×312 \times 8 \times 3 m room.
    4. A fish tank is 40×25×2040 \times 25 \times 20 cm. Find the volume in cm^3 and the capacity in L.
    5. A cube has volume 125125 cm^3. Find the edge length.
    6. A rectangular tank has base 8080 cm by 6060 cm and height 5050 cm. Find the capacity in litres.
Fluency

Triangular prism

    1. A triangular prism has a triangular base of base 66 cm and height 44 cm, and length 1010 cm. Find the volume.
    2. A tent has a triangular cross-section of base 22 m and height 1.51.5 m, and is 2.52.5 m long. Find the volume.
    3. A wedge-shaped doorstop has a right-triangular base with legs 44 cm and 66 cm, and is 88 cm wide. Find the volume.
    4. A triangular prism has volume 120120 cm^3. Its length is 1010 cm. What is the area of the triangular base?
Reasoning

Explain and reason

    1. Ben writes the volume of a 4×4×44 \times 4 \times 4 cube as 42=164^2 = 16 cm^3. What mistake has Ben made?
    2. Explain in your own words why 11 cm^3 =1= 1 mL.
    3. Two rectangular tanks have the same capacity. Must they have the same surface area? Give a reason or a counter-example.
    4. Without calculating, decide which has the greater volume: a cube of side 66 cm, or a rectangular prism of 5×6×75 \times 6 \times 7 cm. Explain briefly.
    5. A rectangular prism and a triangular prism both have length 1010 cm. The rectangular prism has a 66 cm by 44 cm base. What base area would the triangular prism need so that they have the same volume?
Problem solving

Real-world problems

    1. A water tank is 1.21.2 m by 0.80.8 m by 1.51.5 m deep. How many litres when full?
    2. A swimming pool is 1515 m long, 88 m wide, and has a uniform depth of 1.51.5 m. How many kilolitres? At $2.50/kL, what is the cost to fill?
    3. A shoebox is 3333 cm ×\times 2222 cm ×\times 1515 cm. Find the volume in cm^3 and in litres (to 22 dp).
    4. A small aquarium holds 3636 L and has base 6060 cm by 3030 cm. What is the water height?
    5. A 2.52.5 L carton is poured into glasses that hold 250250 mL each. How many full glasses?
    6. A chocolate bar is a triangular prism with equilateral cross-section (side 33 cm, height ≈2.6\approx 2.6 cm) and length 1212 cm. Find its volume (to the nearest cm^3).
Answers

Answer key

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Year 7 core - answers

Fluency

Unit conversions

    1. 2.52.5 L
    2. 46004600 mL
    3. 75007500 mL
    4. 30003000 L
    5. 4.54.5 cm^3
    6. 20002000 cm^3
Fluency

Rectangular prism

    1. 120120 cm^3
    2. 343343 cm^3
    3. 288288 m^3
    4. 20 00020\,000 cm^3 =20= 20 L
    5. 55 cm
    6. 240240 L. Method: V=240 000V = 240\,000 cm^3 =240= 240 L.
Fluency

Triangular prism

    1. 120120 cm^3. Method: base area 12×6×4=12\tfrac{1}{2} \times 6 \times 4 = 12; ×10\times 10.
    2. 3.753.75 m^3. Method: base area 1.51.5; ×2.5\times 2.5.
    3. 9696 cm^3. Method: base area 12×4×6=12\tfrac{1}{2} \times 4 \times 6 = 12; ×8\times 8.
    4. 1212 cm^2. Method: 120÷10120 \div 10.
Reasoning

Explain and reason

    1. Ben used 424^2 (the area of one face) instead of 434^3 (the volume of the cube). Volume of a cube is s3s^3: 43=644^3 = 64 cm^3.
    2. One millilitre of water fills a cube of side 11 cm - this was built into the metric system by definition.
    3. Not necessarily. Example: a 10×10×1010 \times 10 \times 10 cube and a 1×1×10001 \times 1 \times 1000 thin prism both have volume 10001000 units^3 but very different surface areas.
    4. Cube is slightly larger: 63=2166^3 = 216; prism =5×6×7=210= 5 \times 6 \times 7 = 210.
    5. Triangle area needs to be 2424 cm^2. Method: rectangular prism volume =6×4×10=240= 6 \times 4 \times 10 = 240; triangular prism volume =24×10=240= 24 \times 10 = 240.
Problem solving

Real-world problems

    1. 14401440 L. Method: V=1.2×0.8×1.5=1.44V = 1.2 \times 0.8 \times 1.5 = 1.44 m^3 =1440= 1440 L.
    2. 180180 kL; $450. Method: 15×8×1.5=18015 \times 8 \times 1.5 = 180 m^3; ×2.50\times 2.50.
    3. 10 89010\,890 cm^3 =10.89= 10.89 L.
    4. 2020 cm. Method: 36 00036\,000 cm^3 ÷(60×30)\div (60 \times 30).
    5. 1010 glasses.
    6. About 4747 cm^3. Method: triangle area ≈12×3×2.6=3.9\approx \tfrac{1}{2} \times 3 \times 2.6 = 3.9 cm^2; ×12≈46.8\times 12 \approx 46.8.

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