Papy

Volume and Surface Area - Edexcel GCSE Higher Maths

Typically 4-8 marks per paper, concentrated on Papers 2 and 3 · Spec 1MA1

Two spec references share this topic on Edexcel's 1MA1. G16 covers the volume of cuboids and other right prisms, cylinders included, and G17 adds the surface area and volume of spheres, pyramids, cones and composite solids. Prisms and cylinders are common content, so Foundation candidates meet them too. What sits at Higher tier is the second half of G17: the cone with its slant height, the sphere and the hemisphere, the solid built by stacking two shapes, and the frustum, which the specification reaches through composite solids in G17 and the similar-shape work in G19.

Five formulae are printed inside the question that needs them rather than on any sheet: curved surface area of a cone, surface area of a sphere, volume of a sphere, volume of a cone and volume of a pyramid. Separately, the Higher Tier Exam Aid issued with the papers for the 2025, 2026 and 2027 series carries the area of a trapezium, volume of a prism as area of cross section times length, and the circumference and area of a circle. Nothing on either list says that a cylinder is a prism with a circular cross section, gives the curved surface area of a cylinder as 2πrh2\pi rh, or tells you which faces belong in a total surface area, and those are the decisions the marks turn on. Most of this topic lands on Papers 2 and 3, where a calculator is allowed; on Paper 1 the numbers are chosen so the answer can stay as a multiple of π\pi.

A short volume question is worth 2-3 marks, and the method mark is for a correct substitution, so put 13×π×62×10\frac{1}{3} \times \pi \times 6^2 \times 10 on its own line before you touch the calculator. Longer questions run to 4-5 marks and combine shapes: a hemisphere on a cylinder, a metal sphere recast as a cylinder, a container whose height has to be extracted from a stated capacity. Three losses repeat across the series. The diameter goes in where the radius belongs, which multiplies a sphere's volume by eight. A total surface area is written down with one face missing, almost always the circular base of a cone or the flat face of a solid hemisphere. And a value rounded to 3 significant figures halfway through drags the final answer outside the range the mark scheme accepts, so hold the full display and round once.

Worked example

A solid is made from a cylinder and a hemisphere. The cylinder has radius 5 cm and height 12 cm. The hemisphere has radius 5 cm, and its flat circular face exactly covers the top of the cylinder. The bottom of the cylinder is a flat circular face. Work out the total surface area of the solid. Give your answer in cm2^2, correct to 3 significant figures.
[5 marks]
  1. The flat base is a circle of radius 5 cm: area =π×52=25π= \pi \times 5^2 = 25\pi cm2^2.
    B1: the base counts, but the top of the cylinder does not, because the hemisphere covers it
  2. Curved surface of the cylinder: 2πrh=2×π×5×12=120π2\pi rh = 2 \times \pi \times 5 \times 12 = 120\pi cm2^2.
    M1: correct substitution into 2πrh2\pi rh, which is not printed anywhere in the paper
  3. Curved surface of the hemisphere is half of a sphere: 12×4πr2=2×π×52=50π\frac{1}{2} \times 4\pi r^2 = 2 \times \pi \times 5^2 = 50\pi cm2^2.
    M1: halving the given 4πr24\pi r^2; the hemisphere's own flat face is joined on, so it is not a surface
  4. Add the three parts: 25π+120π+50π=195π25\pi + 120\pi + 50\pi = 195\pi cm2^2.
    M1: all three areas summed, with no fourth face and none counted twice
  5. 195π=612.61…195\pi = 612.61\ldots, so the total surface area is 613 cm2^2 to 3 significant figures.
    A1: rounding done once, at the end; 612 here is a truncation, not a rounding

Practice questions

These are original questions written in Edexcel 1MA1 Higher style. There are no diagrams, so every solid is described in full: check whether a length is a radius or a diameter, and whether a height is perpendicular or slant, before you substitute anything. Where a question asks for an answer in the form kπk\pi, type the value of kk only, and where a rounding is stated, keep the unrounded value on your calculator until the final line.

No account needed: the questions below are always free to check, plus 8 extra questions today. Your score so far updates as you go.

  1. Question 1 Exam pace [4 marks]
    A solid cylinder has radius 5 cm and height 8 cm. Its volume is kπk\pi cm3^3 and its total surface area is mπm\pi cm2^2. Write down the values of kk and mm.
  2. Question 2 Exam pace [2 marks]
    A solid hemisphere has radius 6 cm. The volume of the hemisphere is kπk\pi cm3^3. Write down the value of kk.
  3. Question 3 Stretch [4 marks]
    A solid is made from a cylinder with a hemisphere on top. The cylinder has radius 3 cm and height 10 cm. The hemisphere has radius 3 cm and its flat face exactly covers the top of the cylinder. The volume of the solid is kπk\pi cm3^3. Write down the value of kk.
  4. Question 4 Stretch [4 marks]
    A cylindrical container has radius 6 cm. It must hold exactly 2 litres when full. 1 litre = 1000 cm3^3. Work out the height of the container in cm, correct to 1 decimal place.
  5. Question 5 Stretch [5 marks]
    A solid is made from a cylinder with a cone on top. The cylinder has radius 6 cm and height 10 cm. The cone has base radius 6 cm and perpendicular height 8 cm, and its base exactly covers the top of the cylinder. The bottom of the cylinder is a flat circular face. The total surface area of the solid is kπk\pi cm2^2. Write down the value of kk.
  6. Question 6 Stretch [5 marks]
    A cone has base radius 10 cm and perpendicular height 24 cm. A cut is made parallel to the base and the small cone on top is removed, leaving a frustum of perpendicular height 18 cm. Work out the volume of the frustum in cm3^3, correct to the nearest whole number.

Common mistakes examiners see

  • Substituting the diameter where the formula wants the radius. A sphere of diameter 12 cm gives 43π×123=2304π\frac{4}{3}\pi \times 12^3 = 2304\pi instead of 43π×63=288π\frac{4}{3}\pi \times 6^3 = 288\pi, eight times too large.

    Before you substitute, write r=…r = \ldots on its own line and halve any diameter there. A volume that comes out roughly eight times the size you expected is a diameter error every time.

  • Dropping the third from a cone or a pyramid: for a cone of radius 6 cm and height 10 cm, giving π×62×10=360π\pi \times 6^2 \times 10 = 360\pi, which is the cylinder around it rather than the cone.

    Say what the shape sits inside. A cone is a third of the cylinder with the same base and height; a pyramid is a third of the prism on the same base. If your cone volume is not smaller than the cylinder's, the third has gone missing.

  • Putting the perpendicular height into πrl\pi r l. For a cone of radius 5 cm and height 12 cm, π×5×12=60π\pi \times 5 \times 12 = 60\pi is not the curved surface area; the slant height is 13 cm, so it should be 65π65\pi.

    The letter ll in the printed formula is the slant height, the sloping distance from the rim to the apex. Get it from Pythagoras first: l2=r2+h2l^2 = r^2 + h^2, so l=25+144=13l = \sqrt{25 + 144} = 13.

  • Leaving a face out of a total surface area, or counting a hidden one. A solid hemisphere of radius 4 cm answered as 2πr2=32π2\pi r^2 = 32\pi has lost the flat circle; the total is 3πr2=48π3\pi r^2 = 48\pi.

    List the faces before you calculate, and mark each one seen or hidden. Where two solids are joined, the joining circle disappears from both, which is why a hemisphere on a cylinder has three surfaces and not five.

  • Mishandling the units at the last step: dividing a volume in cm3^3 by 100 to reach litres, or treating 1 m3^3 as 100 cm3^3.

    Learn the two conversions these questions rely on: 1 litre is 1000 cm3^3, and 1 m3^3 is 1 000 000 cm3^3, because each metre carries a factor of 100 and the length is cubed. Then check the answer is sensible, since a household water butt holds a couple of hundred litres, not a couple of hundred thousand.

Frequently asked questions

Are the volume of a cone and sphere formulae given in the Edexcel GCSE maths exam?
Yes, but not on a separate sheet. Curved surface area of a cone, surface area of a sphere, volume of a sphere and volume of a cone are printed inside the body of the question that needs them, which is how the 1MA1 specification lists them. For the 2025, 2026 and 2027 series the volume of a pyramid is supplied the same way, so five formulae in this topic arrive with the question rather than from memory.
Do I still have to learn the volume of a cylinder?
Yes. The Higher Tier Exam Aid gives volume of a prism as area of cross section times length and area of a circle as πr2\pi r^2, and a cylinder is a prism with a circular cross section, so the two combine to πr2h\pi r^2 h. The curved surface area of a cylinder, 2πrh2\pi rh, is not given anywhere, and neither is the total surface area, so both are worth knowing on sight.
Are frustums on the Edexcel GCSE Higher specification?
Frustums are examined through G17, which asks for the surface area and volume of composite solids. A frustum is a cone or pyramid with a smaller, mathematically similar cone or pyramid cut off the top, so the method is always the volume of the whole minus the volume of the piece removed. Finding the small radius uses the similar-shapes work in G19.
How many marks is volume and surface area worth on Edexcel Higher?
A single prism, cylinder or sphere calculation is usually 2-3 marks in the first half of a paper. A compound solid, a frustum, or a question that works backwards from a volume to a missing length is typically 4-5 marks later on. Across a full paper of 80 marks expect roughly 4-8 marks, mostly on Papers 2 and 3.

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