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This page sets out the method for cylinders in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Cylinders topic page.
When you need how much a cylinder holds or how much material it is made of.
When the question is about a label, painting, or covering a cylinder.
Volume = πr²h.
Square the radius, multiply by the height, and keep π if an exact answer is wanted.
If you are given the diameter, halve it before you square.
Using the diameter in r² makes the volume four times too big.
The curved surface unrolls to a rectangle of width 2πr and height h.
Its area is 2πrh.
Add the curved surface and the two circular ends.
Total = 2πrh + 2πr².
The question often says “give your answer in terms of π”.
The number in front of π is what you calculate. Do not replace π with 3.14 unless asked.
Volume in cm³, surface area in cm².
A height in metres and a radius in centimetres must be converted first.
Look at the diagram before you substitute. The formulas square the radius and they multiply by the radius. If a tin has diameter 8 cm, the radius is 4 cm. Squaring the diameter instead makes the volume four times too big, because (2r)^2 = 4r^2. Write r = ldots on its own line so the substitution is easy to check.
Use the π key on the calculator. Do not type 3.14 unless the question tells you to use that value. Keep the full calculator display while you work, and round only the final answer. If you need both a volume and an area, round each one at the end from the full value. Do not feed a rounded volume back into another calculation.
Method: 1. Halve the diameter before you substitute, and write r on its own line. 2. Diameter 8cm means r=4cm. Squaring 8 instead multiplies the volume by 4. 3. Use the π key, and round only the final answer.
A cylinder is a prism, so its volume is the area of the cross-section times the length. The cross-section is a circle, and the length is the height of the cylinder. Area of the circle is π r^2. Multiply by the height: V = π r^2 h. For r = 4 cm and h = 10 cm that is π × 16 × 10 = 160π.
The unit is cubic. If r and h are in centimetres, the volume is in cm^3. To the nearest whole number, 160π = 502.654ldots becomes 503 cm^3, because the first decimal digit is 6. Quote the rounded value and the unit. An answer of 503 with no unit can lose the final mark even when the arithmetic is right.
Method: 1. Multiply the circle area by the height: V=π r^2h. 2. For r=4cm and h=10cm, work out π× 16× 10=160π. 3. Round 160π to 503cm^3.
Imagine cutting the label off a tin and laying it flat. The label was wrapped once around the circle, so one side of the rectangle equals the circumference 2π r. The other side equals the height h. The area of that rectangle is 2π r h. That is the curved surface, and it does not include the top or the base.
A cylinder with radius 3 cm and height 8 cm has curved surface 2 × π × 3 × 8 = 48π. The calculator shows 150.796ldots. To 1 decimal place, look at the second decimal digit, which is 9, so 150.7 rounds up to 150.8 cm^2. Notice you do not square the radius here. Squaring belongs to the area of a circle, not to the label.
Method: 1. Treat the label as a rectangle of sides 2π r and h. 2. The curved area is 2π rh. 3. Radius 3cm and height 8cm give 48π=150.8cm^2 to 1 decimal place.
A closed cylinder has three pieces of surface: the curved side and two identical circles. Each circle has area π r^2, so the two ends contribute 2π r^2. Add the curved surface: 2π r^2 + 2π r h. You can factor this as 2π r(r + h), which is a useful check. Both forms must give the same number.
For radius 5 cm and height 12 cm the two ends are 2π × 25 = 50π and the curved surface is 2π × 5 × 12 = 120π. The total is 170π = 534.070ldots, which is 534.1 cm^2 to 1 decimal place. The volume of the same cylinder is 300π = 942.477ldots, which is 942 cm^3 to the nearest whole number. Volume and area are different quantities, so they keep different units and they are rounded separately.
Method: 1. Add the curved surface to two circles: 2π r^2+2π rh. 2. The factored check is 2π r(r+h). 3. Radius 5cm and height 12cm give 50π+120π=534.1cm^2 to 1 decimal place.
A step up in the exam is to be given the volume and asked for a missing length. The formula V = π r^2 h is just an equation with one unknown, so you rearrange it. For a missing height, h = V ÷ (π r^2). For a missing radius, r^2 = V ÷ (π h), and then you take a square root as the final step.
Watch the order of operations on the calculator. Divide by π r^2 as one block, using brackets: V ÷ (π × r^2). A very common slip is to type V ÷ π × r^2, which multiplies by r^2 instead. If a diameter is given, halve it first, and always say what you are finding before you substitute, for example "Height = ldots".
Method: 1. Write V = π r^2 h and substitute the numbers you know. 2. Rearrange to get the unknown on its own, dividing by the whole product π r^2 or π h. 3. Enter the division with brackets, then take a square root if you are finding a radius. 4. Round at the end and check by substituting back into the volume formula.
Tip: Check your answer by putting it back into π r^2 h. It should return the volume you started with, to the rounding you used.
Real questions often use a cylinder as a tank, a pipe or a tin, and then ask for litres. A litre is 1000 cm^3, so divide a volume in cubic centimetres by 1000. If the lengths are in metres, then 1 m^3 = 1000 litres, so multiply instead. Decide which unit your volume is in before you convert.
Flow questions treat the water in a pipe as a long cylinder. In one second the water moving along the pipe is a cylinder of the pipe radius and a height equal to the speed. So the volume per second is π r^2 × speed. Multiply by 60 for a minute, then convert to litres. Time to fill a tank is the volume of the tank divided by the volume per minute.
Method: 1. Make all the lengths use the same unit before you start. 2. Find the volume with π r^2 h, keeping the full calculator value. 3. Convert: cm^3 ÷ 1000 for litres, or m^3 × 1000 for litres. 4. For flow, find the volume per second first, then scale up to the time asked.
Exam questions rarely ask for just a plain closed cylinder. An open tin, or a water butt with no lid, has only one circular end, so the surface area is π r^2 + 2π r h. A pipe is a hollow cylinder, and its volume is the large cylinder minus the hole: π R^2 h - π r^2 h = π (R^2 - r^2) h. Read the wording carefully to see which faces count.
Paint questions use the surface area, and cost questions multiply by a price per tin or per square metre. A final step is often a comparison such as "Is one tin of paint enough?". Show the area, the amount of paint available, and a clear sentence. Round up when you are buying whole tins, because you cannot buy part of a tin.
Method: 1. List which faces are painted or filled: curved surface, one circle, or two circles. 2. Use 2π r h for the curved part and π r^2 for each circular end. 3. For a hollow cylinder, subtract the volume of the hole from the outer volume. 4. Compare with the amount available, then round up for whole tins or items.
Tip: Write a one line plan before calculating, such as "curved surface plus one end", so you do not forget a face.
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