A prism has the same cross-section all along its length, so its volume is that area times the length. A cylinder is a prism with a circular cross-section, and its curved surface unrolls into a rectangle.
Volume of a prism \(= \text{cross-section area} \times \text{length}\) (\(V = Al\)), so a cylinder has \(V = \pi r^2 h\). Its curved surface area is \(2\pi rh\). All three are in the List of formulas; the TOTAL surface area of a cylinder, \(2\pi rh + 2\pi r^2\), is not, so add the two circular ends yourself.
Fig. 1: the net of a cylinder of radius 3 cm and height 8 cm, drawn to scale; the curved surface unrolls to a 6π cm by 8 cm rectangle
Section 01
Prisms and cuboids.
In the lab, slide the cutting plane along the prism: every slice has the same shape and area. That is what makes it a prism, and why the volume is simply the cross-section area times the length.
Volume of a prism \(V = Al\), \(A\) = area of the cross-section, \(l\) = length (given in the List of formulas).
A cuboid is a prism with a rectangular cross-section: \(V = lwh\).
Surface area = total area of all the faces. Sketch the net, or list the faces in pairs.
"Prism" means ANY solid with a uniform cross-section, including half-cylinders and cylindrical sectors.
Worked example Cuboid
A cuboid measures 8 cm by 5 cm by 3 cm. Find its volume and its surface area.
Volume \(= 8 \times 5 \times 3 = 120\).
Faces come in three pairs: \(8 \times 5 = 40\), \(8 \times 3 = 24\), \(5 \times 3 = 15\).
Surface area \(= 2(40 + 24 + 15) = 158\).
Answer: Volume \(120\text{ cm}^3\), surface area \(158\text{ cm}^2\)
Worked example Triangular prism
A prism is 15 cm long. Its cross-section is a right-angled triangle with sides 6 cm, 8 cm and 10 cm (the right angle is between the 6 cm and 8 cm sides). Find its volume and surface area.
Three rectangles, each 15 cm long: \((6 + 8 + 10) \times 15 = 360\).
Surface area \(= 48 + 360 = 408\).
Answer: Volume \(360\text{ cm}^3\), surface area \(408\text{ cm}^2\)
Examiner trap
Surface area means EVERY face. In a triangular prism it is easy to count the two triangles and forget one of the three rectangles, especially the sloping one.
Section 02
Cylinders, unrolled.
In the lab, press Unroll: the curved surface opens out into a rectangle. Its height is the cylinder's height and its width is the circumference \(2\pi r\), so its area is \(2\pi rh\) (Fig. 1).
Worked example Volume and surface area
A cylinder has radius 3 cm and height 8 cm. Find (a) its volume (b) its total surface area. Give your answers in terms of \(\pi\).
The List of formulas gives the CURVED surface area \(2\pi rh\) only. For a closed cylinder add \(2\pi r^2\) for the ends; for an open tin (no lid) add just one \(\pi r^2\).
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
Which cylinder formula is NOT printed in the List of formulas?
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A prism has cross-section area \(12\text{ cm}^2\) and length 9 cm. What is its volume?
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Find the volume of a cylinder of radius 2 cm and height 5 cm, in terms of \(\pi\).
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How many faces does a cuboid have?
Section 03
Working backwards.
If the volume is given, put it into the formula and solve for the missing length. Keep full calculator values until the last step.
Worked example Height of a cylinder
A cylinder has volume \(500\text{ cm}^3\) and radius 5 cm. Find its height.
\(\pi \times 5^2 \times h = 500\), so \(25\pi h = 500\).
\(h = \dfrac{500}{25\pi} = 6.366...\)
Answer: 6.37 cm
Worked example A half-cylinder prism
A prism is 20 cm long and its cross-section is a semicircle of radius 4 cm. Find its volume.
Water fills a cuboid 20 cm by 15 cm by 12 cm. It is poured into an empty cylinder of radius 8 cm. How deep is the water in the cylinder?
Volume of water \(= 20 \times 15 \times 12 = 3600\text{ cm}^3\).
In the cylinder: \(\pi \times 8^2 \times h = 3600\).
\(h = \dfrac{3600}{64\pi} = 17.90...\)
Answer: 17.9 cm
Examiner trap
Rounding the radius or an area too early changes the third significant figure. Store values in the calculator and round once at the end.
Stage 2 · Exam
Exam-style questions
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A closed cylinder has radius 5 cm and height 12 cm. Find its total surface area.
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Water fills a cuboid tank 25 cm by 16 cm by 10 cm. All the water is poured into an empty cylinder of radius 9 cm. Find the depth of water in the cylinder.
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A prism is 15 cm long. Its cross-section is a trapezium with parallel sides 5 cm and 9 cm, 4 cm apart. Find its volume.
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An open tin (no lid) is a cylinder of radius 4 cm and height 10 cm. Find the area of metal used to make it.
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Prisms and cylinders, covered.
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