Study Companion
Topic 09 | Measurement & Space

Volume of right prisms

Year 8 core: volumes and capacities of right prisms (rectangular, triangular, and general prisms), and problems involving rates of flow.

45-60 min Printable practice Answer key Challenge included
VC2M8M02
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What you will learn

1. The key idea: base area × length

Volume of a right prism
V=Abase×L,V = A_{\text{base}} \times L,

where AbaseA_{\text{base}} is the area of the prism’s cross-section (a constant shape along its length), and LL is the length (the distance between the two matching ends).

rectangulartriangulartrapezoidal
Three right prisms. The shaded end is the cross-section (the base). Volume = cross-section area × length.
Worked example 1 Rectangular prism

A box is 88 cm by 55 cm by 44 cm.

V=8×5×4=160 cm3.V = 8 \times 5 \times 4 = 160 \text{ cm}^3.
Worked example 2 Triangular prism

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

  1. Triangle area: 12×2.4×1.8=2.16\tfrac{1}{2} \times 2.4 \times 1.8 = 2.16 m2^2.
  2. Volume: 2.16×3=6.482.16 \times 3 = 6.48 m3^3.
Worked example 3 Trapezoidal prism

A channel has a trapezoidal cross-section: parallel sides 1.51.5 m and 2.52.5 m, height 0.80.8 m. It is 2020 m long.

  1. Trapezium area: 12(1.5+2.5)×0.8=1.6\tfrac{1}{2}(1.5 + 2.5) \times 0.8 = 1.6 m2^2.
  2. Volume: 1.6×20=321.6 \times 20 = 32 m3^3.

2. Volume ↔ capacity

Conversions

Volume units
1 cm3=1000 mm3,1 m3=1 000 000 cm3.1 \text{ cm}^3 = 1000 \text{ mm}^3, \qquad 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}, \qquad 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}.

3. Rates and time to fill

If a volume VV is delivered at a constant rate rr, the time to fill is t=Vrt = \dfrac{V}{r}.

Worked example 4 Filling a tank

A rectangular tank is 22 m by 1.51.5 m by 1.21.2 m deep. A hose delivers 1515 L per minute. How long to fill the tank?

  1. Volume: V=2×1.5×1.2=3.6V = 2 \times 1.5 \times 1.2 = 3.6 m3^3 =3600= 3600 L.
  2. Time: t=360015=240t = \dfrac{3600}{15} = 240 min =4= 4 hours.

Practice: Year 8 core

Fluency

Volume of right prisms

    1. Find the volume of a cuboid 7×4×37 \times 4 \times 3 cm.
    2. Find the volume of a cube of side 66 cm.
    3. Find the volume of a triangular prism with triangle base 1010 cm, height 66 cm, and length 1515 cm.
    4. Find the volume of a trapezoidal prism: parallel sides 44 m and 66 m, height 33 m, length 88 m.
    5. A cuboid has volume 480480 cm3^3 and a base of 8×58 \times 5 cm. Find its height.
    6. A triangular prism has volume 150150 cm3^3 and length 1010 cm. Find the area of its triangular base.
Fluency

Volume and capacity

    1. Convert 35003500 mL to L.
    2. Convert 4.24.2 L to mL.
    3. Convert 0.750.75 m3^3 to L.
    4. Convert 23002300 cm3^3 to mL.
    5. A fish tank is 60×30×4060 \times 30 \times 40 cm. What is its capacity in L?
    6. A pool is 1212 m by 66 m by 1.51.5 m. How many kL?
Fluency

Rates and time

    1. A 300300 L tank fills at 55 L/min. How long?
    2. A hose delivers 0.20.2 L/s. How long (minutes) to fill a 6060 L drum?
    3. A pool of 90 00090\,000 L is filled at 150150 L/min. How many hours?
    4. A dripping tap loses 44 drops/sec and 2020 drops = 11 mL. How much water in a day (L)?
Reasoning

Explain and spot the mistake

    1. Liam writes the volume of a 22 cm cube as 88 cm2^2. What is the error?
    2. Two cuboids have the same volume. Must they have the same surface area? Give a reason.
    3. Explain why 11 cm3=1^3 = 1 mL using the definition of the metric system.
    4. A tap flows at 1212 L/min. Without calculating, decide whether a 10001000 L tank fills in under or over 11 hour. Justify.
Problem solving

Real contexts

    1. A water tank is 1.21.2 m by 0.80.8 m by 1.51.5 m deep. A hose delivers 2020 L/min. How long to fill?
    2. A swimming pool is 1515 m by 88 m with uniform depth 1.41.4 m. If water costs $2.40/kL, find the total fill cost.
    3. A shoebox is 3333 cm by 2222 cm by 1515 cm. Give its volume in cm3^3 and in litres (to 2 dp).
    4. A chocolate bar in the shape of a triangular prism has equilateral cross-section (side 33 cm, approximate height 2.62.6 cm) and length 1010 cm. Find its volume (to the nearest cm3^3).

Challenge

Reasoning

Harder problems

    1. A cube has surface area 150150 cm2^2. Find its volume.
    2. Two identical right-angled triangular prisms are joined along their rectangular faces to form a cuboid. If each prism has legs 33 cm and 44 cm and length 1010 cm, find the volume of the cuboid.
    3. A rectangular tank (22 m × 11 m × 11 m) is being filled at 2525 L/min while draining at 1010 L/min. How long to fill if both taps are open?
    4. A glass is a cylinder of radius 33 cm and height 1010 cm. Find its capacity in mL. (Use V=πr2hV = \pi r^2 h; π≈3.14\pi \approx 3.14.)
Answers

Answer key

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Year 8 core - answers

Fluency

Volume of right prisms

    1. 8484 cm3^3
    2. 216216 cm3^3
    3. 450450 cm3^3. Method: triangle area 3030; × 1515.
    4. 120120 m3^3. Method: trapezium area 12(4+6)×3=15\tfrac{1}{2}(4+6) \times 3 = 15; × 88.
    5. 1212 cm. Method: 480÷40480 \div 40.
    6. 1515 cm2^2. Method: 150÷10150 \div 10.
Fluency

Volume and capacity

    1. 3.53.5 L
    2. 42004200 mL
    3. 750750 L
    4. 23002300 mL
    5. 7272 L. Method: V=72 000V = 72\,000 cm3=72^3 = 72 L.
    6. 108108 kL. Method: V=108V = 108 m3^3.
Fluency

Rates and time

    1. 6060 min =1= 1 h.
    2. 55 min. Method: 0.20.2 L/s =12= 12 L/min; 60÷1260 \div 12.
    3. 1010 hours. Method: 90 000÷150=60090\,000 \div 150 = 600 min.
    4. About 17.317.3 L/day. Method: 4×86 400=345 6004 \times 86\,400 = 345\,600 drops/day; ÷20=17 280\div 20 = 17\,280 mL.
Reasoning

Explain and spot the mistake

    1. 88 is correct numerically but the units should be cm3^3 (cube has volume in cubic units). cm2^2 is area.
    2. Not necessarily. Example: 1×1×10001 \times 1 \times 1000 and 10×10×1010 \times 10 \times 10 both have volume 10001000 but very different surface areas.
    3. The metric system was built so that 11 cm3^3 holds exactly 11 mL of water. They are different units measuring the same amount of space or liquid.
    4. Over 11 hour? 1212 L/min × 6060 = 720720 L in an hour - only 720720 of 10001000 L. So it takes over 11 hour.
Problem solving

Real contexts

    1. 7272 min. Method: V=1.44V = 1.44 m3=1440^3 = 1440 L; 1440÷201440 \div 20.
    2. $403.20. Method: V=15×8×1.4=168V = 15 \times 8 \times 1.4 = 168 m3^3 = 168168 kL; × 2.402.40.
    3. 10 89010\,890 cm3=10.89^3 = 10.89 L.
    4. 3939 cm3^3. Method: triangle area 12×3×2.6=3.9\tfrac{1}{2} \times 3 \times 2.6 = 3.9; × 10=3910 = 39.

Challenge - answers

Reasoning

Harder problems

    1. 125125 cm3^3. Method: 6s2=1506 s^2 = 150, so s2=25s^2 = 25, s=5s = 5; V=53V = 5^3.
    2. 120120 cm3^3. Method: two prisms combine so half of the cuboid; each triangle area 66; cuboid =4×3×10=120= 4 \times 3 \times 10 = 120.
    3. 133.3133.3 min (about 22 h 1313 min). Method: V=2V = 2 m3=2000^3 = 2000 L; net fill rate =15= 15 L/min; 2000÷152000 \div 15.
    4. 282.6282.6 mL (approx). Method: V=π×9×10≈282.6V = \pi \times 9 \times 10 \approx 282.6 cm3^3.

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