4.10 - Prisms, cylinders and volume units

Find the space inside prisms and cylinders, choose which cylinder surfaces to count, and convert volumes into capacities. These methods help you decide how much a container holds and how much material covers it.

Volume of a prism

Can a tank with internal dimensions $0.8\,\mathrm{m}$ , $0.5\,\mathrm{m}$ and $0.4\,\mathrm{m}$ hold $150$ litres? First we need a way to measure the space inside it.

You already know

A rectangle has area $wh$ ; a triangle has area $\frac12 bh$ , using perpendicular height. Multiplying three lengths gives cubic units.

Start with a cuboid measuring $8\,\mathrm{cm}$ by $3\,\mathrm{cm}$ by $2\,\mathrm{cm}$ . Each $1\,\mathrm{cm}$ -thick layer contains $3\times2=6$ unit cubes. Eight layers contain $6\times8=48$ unit cubes, so the volume is $48\,\mathrm{cm}^3$ .

Prism and cross-section

A prism has two congruent, parallel polygonal ends and an unchanged cross-section along its length. A cross-section is the shape of a cut perpendicular to that length. A triangular prism is an example; a cone tapers, so it is not a prism.

The stacking argument works for any cross-section with a known area. Use the distance between the identical ends, rather than a sloping edge of an end face.

Volume of a prism

V=A\ell

Here $A$ is the cross-sectional area and $\ell$ is the length between the ends. On both Foundation and Higher sheets: volume equals area of cross-section times length. In the formulae sheet.

The shaded triangle below is one whole end. Its two perpendicular dimensions find the area; the separate $9\,\mathrm{cm}$ dimension supplies the prism length.

[DIAGRAM: asset_slug: m078_prism; caption: A triangular cross-section and the separate prism length; description: Schematic triangular prism with right-triangular cross-section base 6 cm and perpendicular height 4 cm. Extrusion between triangular ends is 9 cm. The three labels describe distinct dimensions.; file: m078_prism.png; recommended_method: matplotlib]
A triangular cross-section and the separate prism length

Worked example 1: a triangular prism

Question. A triangular prism has end-face base $6\,\mathrm{cm}$ , perpendicular height $4\,\mathrm{cm}$ , and prism length $9\,\mathrm{cm}$ . Find its volume.

Plan. Find one triangular end's area, then multiply by the prism length.

\begin{aligned} A&=\frac12\times6\times4=12\,\mathrm{cm}^2 &&\text{triangle area}\\ V&=12\times9=108\,\mathrm{cm}^3 &&\text{stack along the length} \end{aligned}

Why this step? Every slice has the same triangular area, so the full length multiplies that area.

Answer. $108\,\mathrm{cm}^3$ .

Check. The triangle is half a $6$ by $4$ rectangle, so the volume is half $6\times4\times9=216$ , as required.

Now change only the end-face shape: a trapezium replaces the triangle inside the same base, height and prism length. The area calculation changes; the final multiplication does not.

Worked example 2: a trapezium end

Question. A prism has a trapezium end with parallel sides $2\,\mathrm{cm}$ and $6\,\mathrm{cm}$ , perpendicular separation $4\,\mathrm{cm}$ , and prism length $9\,\mathrm{cm}$ . Find its volume.

Plan. Use the trapezium-area formula, then multiply by $9$ .

\begin{aligned} A&=\frac12(2+6)\times4=16\,\mathrm{cm}^2 &&\text{trapezium area}\\ V&=16\times9=144\,\mathrm{cm}^3 &&\text{area times length} \end{aligned}

Why this step? The parallel sides describe the end face, so neither is the prism length.

Answer. $144\,\mathrm{cm}^3$ .

Check. Its area lies between the enclosing rectangle's $24\,\mathrm{cm}^2$ and the triangle's $12\,\mathrm{cm}^2$ ; its volume lies between $216$ and $108\,\mathrm{cm}^3$ .

A formula can be supplied and still need careful interpretation. Identify the cross-section before choosing its dimensions.

Watch out

The line $V=6\times4\times9=216$ is wrong for the triangular prism: it uses a rectangular end. Correct it to $V=\frac12\times6\times4\times9=108\,\mathrm{cm}^3$ .

Try changing just the shorter parallel side in the trapezium example.

Volume of a cylinder

A cylinder has the same circular cross-section all the way along its perpendicular height. It has curved sides, so we do not call it a polygonal prism, but the same stacking argument gives its volume.

A circle of radius $3\,\mathrm{cm}$ has area $9\pi\,\mathrm{cm}^2$ . Extending it through $10\,\mathrm{cm}$ gives $90\pi\,\mathrm{cm}^3$ . Radius means centre to rim; diameter means the full distance across the circle through its centre.

Volume of a cylinder

V=\pi r^2h

Use radius $r$ and perpendicular height $h$ in the same length unit. This result is supplied on both Foundation and Higher sheets. In the formulae sheet.

Circle area $\pi r^2$ and the relationship $d=2r$ are not printed on either tier sheet: learn them. For capacity, use internal dimensions; the outside includes the container's walls.

Worked example 3: radius given

Question. A cylinder has radius $3\,\mathrm{cm}$ and height $10\,\mathrm{cm}$ . Find its volume to three significant figures.

Plan. Substitute the radius and height; keep $\pi$ until rounding.

\begin{aligned} V&=\pi\times3^2\times10 &&\text{substitute}\\ &=90\pi\,\mathrm{cm}^3 &&\text{exact volume}\\ &\approx283\,\mathrm{cm}^3 &&\text{three significant figures} \end{aligned}

Why this step? The squared quantity is the radius because $\pi r^2$ is the circular area.

Answer. $283\,\mathrm{cm}^3$ .

Check. It fits inside a $6$ by $6$ by $10$ cuboid, whose volume is $360\,\mathrm{cm}^3$ , so the answer is sensibly smaller.

Change only how the circle's size is supplied: give the diameter of the same cylinder. The geometry and volume stay unchanged.

Worked example 4: diameter given

Question. A cylinder has diameter $6\,\mathrm{cm}$ and height $10\,\mathrm{cm}$ . Find its volume to three significant figures.

Plan. Halve the diameter before using the volume formula.

\begin{aligned} r&=6\div2=3\,\mathrm{cm} &&\text{radius is half the diameter}\\ V&=\pi\times3^2\times10=90\pi\,\mathrm{cm}^3 &&\text{substitute the radius}\\ V&\approx283\,\mathrm{cm}^3 &&\text{round at the end} \end{aligned}

Why this step? A diameter is twice the radius; entering it as $r$ would square an extra factor of $2$ .

Answer. $283\,\mathrm{cm}^3$ .

Check. This is the same physical cylinder as in Worked example 3, so its volume must agree.

You can also work backwards: if volume and radius are known, divide by the circular area to find $h=V/(\pi r^2)$ . The supplied formula is useful only when each symbol has been matched to the right measurement.

Watch out

$V=\pi\times6^2\times10$ is wrong when $6\,\mathrm{cm}$ is the diameter. Use $r=3\,\mathrm{cm}$ ; the wrong line gives four times the correct volume.

For the next cylinder, change only the diameter; the height stays the same.

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Number

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Handling data

4.10 - Prisms, cylinders and volume units

Volume of a prism

Volume of a cylinder

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