Secondary 3 Volume of Solids Worksheet

All 14 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1The Area and the Volume of Solids

   A juice shop uses closed cylindrical tins with a radius of $10\text{ cm}$ and a height of $20\text{ cm}$. For each task below, say whether it needs a length, an area or a volume, give the unit that answer would carry, then compute it. Use $\pi$ on your calculator and round to two decimal places.

   1. Printing a paper label that wraps once around the curved side, top to bottom, with no overlap.
   2. Filling the tin with juice.
   3. Tying a ribbon once around the rim of the lid.
2. Q2The Volume of Cones

   Find the volume of each cone. Round to two decimal places.

   1. A paper snow-cone cup with a radius of $4\text{ cm}$ and a height of $9\text{ cm}$.
   2. A pile of road salt shaped like a cone, $12\text{ m}$ across at the base and $10\text{ m}$ high.
3. Q3The Volume of Cylinders

   A cylindrical compost bin is $60\text{ cm}$ across and $90\text{ cm}$ tall.

   1. Find its volume in ${\text{cm}}^3$, rounded to the nearest ${\text{cm}}^3$.
   2. Knowing that $1\text{ L}=1000{\text{ cm}}^3$, give its capacity in litres, rounded to one decimal place.
4. Q4The Volume of Decomposable Solids

   A grain silo is made of a cylinder of radius $3\text{ m}$ and height $8\text{ m}$, topped by a cone of the same radius whose height is $4\text{ m}$. Find the total volume of the silo, exactly in terms of $\pi$ and then rounded to two decimal places.

5. Q5The Volume of Prisms

   Find the volume of each prism.

   1. A chocolate bar shaped like a prism $15\text{ cm}$ long whose base is a right triangle with legs of $8\text{ cm}$ and $5\text{ cm}$.
   2. A feeding trough shaped like a prism $12\text{ cm}$ long whose base is a trapezoid with parallel sides of $10\text{ cm}$ and $6\text{ cm}$ and a height of $4\text{ cm}$.
6. Q6The Volume of Pyramids

   Find the volume of each pyramid.

   1. A square-based pyramid with a base edge of $9\text{ cm}$ and a height of $10\text{ cm}$.
   2. A pyramid whose base is a rectangle $12\text{ cm}$ by $5\text{ cm}$, with a height of $7\text{ cm}$.
7. Q7The Volume of Solids Using Algebra

   Write each volume as a simplified algebraic expression, then evaluate it for the value given.

   1. A rectangular box whose width is $x$ cm, whose length is $4\text{ cm}$ more than its width, and whose height is $5\text{ cm}$. Evaluate for $x=3$.
   2. A cube whose edge is $2a$ cm. Evaluate for $a=1.5$.
8. Q8The Volume of Truncated Solids

   A concrete planter is a truncated square-based pyramid. It comes from a pyramid with a base edge of $12\text{ cm}$ and a height of $18\text{ cm}$, from which the top pyramid — base edge $4\text{ cm}$, height $6\text{ cm}$ — has been removed. Find the volume of the planter.

9. Q9The Volume of a Cube

   Answer both parts.

   1. A wooden block is a cube of edge $7\text{ cm}$. Find its volume.
   2. Another cube has a volume of $512{\text{ cm}}^3$. Find its edge, then its total surface area.
10. Q10The Volume of a Sphere

    Find the volume of each sphere, rounded to two decimal places.

    1. A ball with a radius of $6\text{ cm}$.
    2. An exercise ball with a diameter of $20\text{ cm}$.
11. Q11Units for Measuring Volume and their Conversion

    Carry out each conversion, and next to each one write the factor you used and where it comes from.

    1. $3.5{\text{ m}}^3$ into ${\text{cm}}^3$.
    2. $45000{\text{ mm}}^3$ into ${\text{cm}}^3$.
    3. $0.02{\text{ m}}^3$ into ${\text{dm}}^3$.
12. Q12Units of Capacity and Their Conversion

    Answer each part, using $1\text{ L}=1000\text{ mL}=1000{\text{ cm}}^3$.

    1. Convert $2.5\text{ L}$ into millilitres.
    2. Convert $3.2\text{ L}$ into ${\text{cm}}^3$.
    3. An aquarium is a rectangular prism $40\text{ cm}$ long, $25\text{ cm}$ wide and $30\text{ cm}$ high. Find its volume in ${\text{cm}}^3$, then its capacity in litres.
13. Q13Units of Time and Their Conversion

    Carry out each conversion.

    1. $2\text{ h }45\text{ min}$ into minutes.
    2. $5400\text{ s}$ into hours and minutes.
    3. An arena's ice resurfacer takes $8\text{ min }30\text{ s}$ per pass. How long do $6$ passes take, in minutes?
14. Q14Synthesis — drawing on several sheets in this topic

    A cylindrical rain barrel has a radius of $30\text{ cm}$ and a height of $80\text{ cm}$.

    1. Find its volume in ${\text{cm}}^3$, rounded to the nearest ${\text{cm}}^3$.
    2. Give its capacity in litres, rounded to one decimal place.
    3. During a storm, water pours in at $4.5\text{ L}/\text{min}$. How long does the barrel take to fill, to the nearest minute? Give the answer in minutes and also in hours and minutes.

The 14 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

Length, area or volume? Ask it first, every time (Q1)

Q1 is the sheet's opening move and its best habit. Three tasks on the same tin — printing a label, filling it with juice, tying a ribbon round the lid — and each one needs a different kind of quantity. Deciding which before choosing a formula does two things at once: it picks the formula for you, and it tells you what unit the answer must carry.

The one third, and where it comes from (Q2, Q4, Q6)

A pyramid and a prism that share a base and a height are not equal: the pyramid holds exactly one third as much. A cone stands in the same relation to a cylinder. So there are really only two formulas on this sheet, each used twice:

Q5 makes the first one concrete with two prisms whose bases are not rectangles — a right triangle and a trapezoid. The formula does not change: find the area of the base, multiply by the length. That works because a prism has the same cross-section all the way along, which is what the word prism means. Q6 does the same for pyramids on a square and on a rectangular base.

The one third is worth feeling rather than memorising. In Q4's silo the cone on top is half as tall as the cylinder below it, yet it contributes only a seventh of the total volume — the third is doing far more work than the height difference suggests.

Read the measurement you were given (Q2, Q10)

Both traps are caught the same way: before writing anything, write down r = … and h = …, in one unit, and only then reach for the formula.

Decomposable and truncated: added or subtracted (Q4, Q8)

These two look similar and are opposite operations. A decomposable solid is built from simple solids, so its volume is the sum — Q4's silo is a cylinder plus a cone, and unlike a surface-area question nothing is hidden, because the join takes away no volume at all. A truncated solid is what remains after a piece is sliced off, so its volume is the whole minus the removed part.

When the removed piece's measurements are not given, similarity supplies them: a cut parallel to the base leaves a small solid similar to the whole one, so its lengths are k times the original's and its volume is k³ times. That is also the fastest check on an answer — if the small solid is a third of the height, it should be about a twenty-seventh of the volume.

Cubic units, litres, and why the factor is cubed (Q11, Q12)

Q11 asks for the conversion factor and where it comes from, because the second part is what makes the first stick. A cubic unit converts three dimensions at once, so the length factor is cubed:

For lengths the step between neighbouring units is 10; for areas it is 10² = 100; for volumes it is 10³ = 1000. Using the length factor on a volume is wrong by ten thousand or a million, which is large enough that a sanity check catches it every time: a cubic metre of gravel is a truckload, so an answer the size of a drinking glass cannot be right.

Capacity is the easy half, and it is worth memorising outright: 1 L = 1000 mL = 1000 cm³, so a millilitre and a cubic centimetre are the same size, and 1 L is exactly 1 dm³. Q3 and Q12 both finish by turning cm³ into litres, which is a single division by 1000 — but only once every length was in centimetres to begin with.

Volume as an expression (Q7)

Q7 writes two volumes algebraically. A box whose length is 4 cm more than its width becomes 5 · x · (x + 4) = 5x² + 20x, and a cube of edge 2a becomes (2a)³ = 8a³ — note that the exponent reaches the 2 as well as the a, which is exactly the step students skip.

Then evaluate, and check against the dimensions themselves. For x = 3 the expression gives 105 cm³, and 3 × 7 × 5 is also 105. That second line is not busywork: it is the only way to discover that the expression, rather than the arithmetic, was where the mistake happened.

Rounding is decided by the situation, not by the decimal (Q14)

Q14 finishes with a rain barrel filling at a steady rate, and asks for a time. A time is rounded to the nearest minute, because the question is how long the filling takes. Compare that with a question asking how many bags of soil to buy, where anything above a whole number rounds up — a bag cannot be bought in pieces, and rounding down leaves the job unfinished.

The same sheet also asks you to move between decimal hours and hours-and-minutes, and there the rule is that time is not decimal: 0.25 h is a quarter of an hour, so 15 minutes, not 25. Keep the whole number as the hours and multiply the decimal part by 60.

Getting the most out of it

Write r, h and the unit before the formula

One line at the top of every question: the radius (halved from the diameter if that is what you were given), the height, and every length converted to a single unit. Most of the errors this sheet is built to catch are already impossible once that line exists.

Say what kind of quantity you are computing

Length, area or volume — and therefore cm, cm² or cm³. The unit is a free check on the formula, and the sheet opens with a question that asks for exactly this because it pays off on every question after it.

Sanity-check the size of the answer

A tank a metre tall does not hold a bottle's worth of water, and a truckload of gravel is not a glass. Every unit-conversion error on this material produces an answer that is wrong by a factor of a hundred or more, so a one-sentence comparison with something familiar catches it before it is written down.

Common questions