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Volume of Solids0%

Measurement & Geometry · Topic 6 of 9

Volume of Solids

Video coming soon3 worked examples

Theory

The Golden Rule: If a volume question asks you to round your final answer to a specific number of significant figures, you must write down your unrounded answer from your calculator display first. Failing to show the unrounded calculation will cost you vital process marks, even if the rounded answer is correct.

1. Using Given Formulae

You do not need to memorise complex volume formulae, as they are provided on the exam paper. You simply need to substitute the correct values into the given equations:

  • Prism: V=AhV = Ah (Area of the base shape × height).
  • Cylinder: V=πr2hV = \pi r^2 h.
  • Sphere: V=43πr3V = \frac{4}{3} \pi r^3.
  • Cone: V=13πr2hV = \frac{1}{3} \pi r^2 h.

2. Hemispheres

To calculate the volume of a hemisphere (half a sphere), simply use the sphere formula provided on the formula sheet and divide your final answer by 2.

3. Composite Solids

Composite solids are made from two or more standard shapes joined together.

  • Identify the basic 3D shapes that make up the solid.
  • Calculate the volume of each part separately.
  • Add the volumes together for the total.

4. Converting to Litres

Volume is often calculated in cubic centimetres (cm3\text{cm}^3), but questions frequently ask for the final answer in litres.

  • 1 litre = 1000 cm3\text{cm}^3.
  • To convert cm3\text{cm}^3 into litres, divide your answer by 1000.

⚠️ Common Examiner Traps

  • The "Radius vs Diameter" Mix-Up: Candidates frequently substitute the full width (diameter) of a circular base directly into the formula. You must always halve the diameter to find the radius (r) before calculating the volume of a cylinder, cone, or sphere.
  • The "Hidden Height" Trap: In composite solids (like a cone sitting on top of a hemisphere), a question will often give you the total height of the object rather than the specific height of the cone. You must subtract the radius of the hemisphere from the total height to find the true height of the cone before using the formula.
  • The "Premature Rounding" Trap: When working out the volume of two separate shapes to add together, candidates sometimes round the volume of the first shape before calculating the second. Keep the exact values in your calculator until the very final step to avoid rounding errors.

Worked examples

Example 1

A child’s toy is in the shape of a hemisphere with a cone on top, as shown in the diagram.

  • The toy is 10 centimetres wide (diameter).
  • The total height of the toy is 16 centimetres.

Calculate the volume of the toy. Give your answer correct to 2 significant figures. (5 marks)

Step 1: Find the radius of the toy by halving the diameter.

10÷2=5 cm radius10 \div 2 = 5 \text{ cm radius}

Step 2: Find the true height of the cone. The 16 cm total is made of the cone's height plus the hemisphere's radius (5 cm).

16−5=11 cm cone height16 - 5 = 11 \text{ cm cone height}

Step 3: Calculate the volume of the cone using V=13πr2hV = \frac{1}{3}\pi r^2 h.

V=13×π×52×11=287.979... cm3V = \frac{1}{3} \times \pi \times 5^2 \times 11 = 287.979... \text{ cm}^3

Step 4: Calculate the volume of the hemisphere using V=43πr3÷2V = \frac{4}{3}\pi r^3 \div 2.

V=43×π×53÷2=261.799... cm3V = \frac{4}{3} \times \pi \times 5^3 \div 2 = 261.799... \text{ cm}^3

Step 5: Add the volumes together and write down the unrounded answer first.

287.979...+261.799...=549.778... cm3287.979... + 261.799... = 549.778... \text{ cm}^3

Step 6: Round to exactly 2 significant figures.

V=550 cm3V = 550 \text{ cm}^3

Example 2

A bottle consists of a cuboid and a cylinder.

  • The cuboid base has dimensions 8 cm by 4.5 cm by 10 cm height.
  • The cylindrical neck has a diameter of 3 cm and a height of 4 cm.

Calculate the total volume of the bottle. (4 marks)

Step 1: Calculate the volume of the cuboid (V=L×B×HV = L \times B \times H).

8×4.5×10=360 cm38 \times 4.5 \times 10 = 360 \text{ cm}^3

Step 2: Find the radius of the cylinder (halve the diameter).

3÷2=1.5 cm3 \div 2 = 1.5 \text{ cm}

Step 3: Calculate the volume of the cylinder using V=πr2hV = \pi r^2 h.

V=π×1.52×4=28.274... cm3V = \pi \times 1.5^2 \times 4 = 28.274... \text{ cm}^3

Step 4: Add the two volumes together.

360+28.274...=388.274...360 + 28.274... = 388.274...

Final Answer: The volume of the bottle is 388.27 cm³ (rounded to 2 d.p.).

Example 3

Dougie is organising a party and will serve juice in cylindrical cups.

  • The cups have a radius of 3.5 cm and a total height of 11 cm.
  • Each cup will be filled with juice to exactly 2 cm from the top.

Calculate the volume of juice in one cup. Give your answer in litres. (4 marks)

Step 1: Determine the actual height of the liquid in the cup.

11 cm−2 cm=9 cm11 \text{ cm} - 2 \text{ cm} = 9 \text{ cm}

Step 2: Calculate the volume of the juice using the cylinder formula.

V=π×3.52×9=346.360... cm3V = \pi \times 3.5^2 \times 9 = 346.360... \text{ cm}^3

Step 3: Convert the cubic centimetres into litres by dividing by 1000.

346.360÷1000=0.34636... litres346.360 \div 1000 = 0.34636... \text{ litres}

Final Answer: There are 0.346 litres of juice in each cup.

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11 Volume of Solids questions with answers and video solutions