Surface area, volume and capacity

Surface area and volume of prisms, cylinders, pyramids, cones and spheres, composite solids, open containers, and converting volume to capacity in litres.

Surface area is how much material it takes to wrap a solid: paint for a water tank, cardboard for a box, steel for a silo. Volume is how much space is inside, and capacity is how much liquid fits in that space. Most real questions want one of these three, so the first job is always to decide which.

Prisms and cylinders

A prism has the same cross-section all the way through, like a loaf of bread. A cylinder is a prism with a circle for its cross-section. For any prism, volume is the area of the end times the length.

Prisms and cylinders

$$V = Ah \qquad V_{\text{cylinder}} = \pi r^2 h \qquad SA_{\text{cylinder}} = 2\pi r^2 + 2\pi r h$$

The curved part of a cylinder unrolls into a rectangle. Its width is the height $h$ and its length is the circumference $2\pi r$, which is where $2\pi r h$ comes from.

Example 1●●

A rainwater tank is a closed cylinder with radius $1.2$ m and height $2$ m. Find (a) its capacity in litres (b) the area of steel used to make it, to two decimal places.

(a) $V = \pi \times 1.2^2 \times 2 \approx 9.0478$ m³

Volume of a cylinder. Keep the full value on the calculator.

Capacity $\approx 9048$ L

1 m³ = 1000 L.

(b) $SA = 2\pi(1.2)^2 + 2\pi(1.2)(2) \approx 24.13$ m²

Two circular ends plus the curved side.

Pyramids and cones

A pyramid or cone holds exactly one third of the prism or cylinder with the same base and height. You can check this with sand and two containers. It takes three cones to fill the cylinder.

Pyramids and cones

$$V_{\text{pyramid}} = \frac13 A h \qquad V_{\text{cone}} = \frac13 \pi r^2 h \qquad SA_{\text{cone}} = \pi r^2 + \pi r l$$

For surface area you need the slant height $l$, measured along the sloping face, not the vertical height $h$. Find it with Pythagoras: $l = \sqrt{r^2 + h^2}$ for a cone.

Example 2●●

A traffic cone (with no base) has radius $15$ cm and height $45$ cm. Find the area of plastic in its curved surface, to the nearest cm².

$l = \sqrt{15^2 + 45^2} = \sqrt{2250} \approx 47.43$ cm

The curved surface uses the slant height, so find it first with Pythagoras.

$SA = \pi \times 15 \times 47.434 \approx 2235$ cm²

No base, so only the curved part πrl. Use the unrounded slant height.

Spheres

Sphere

$$SA = 4\pi r^2 \qquad V = \frac43 \pi r^3$$

Example 3●

A basketball has radius $12$ cm. Find its surface area and the volume of air inside, and give the volume in litres.

$SA = 4\pi \times 12^2 \approx 1809.56$ cm²

Substitute r = 12 into each formula.

$V = \dfrac43 \pi \times 12^3 \approx 7238.23$ cm³

The radius is cubed for volume.

$V \approx 7.24$ L

1000 cm³ = 1 L.

InteractiveSolid builder

Try this

Pick Cylinder. Double the radius. By what factor does the volume grow? Now double the height instead. Explain the difference.
Pick Cone and then Cylinder with the same radius and height. Check that the cone's volume is one third.
Pick Box and turn on Open top. Which face disappears from the surface area?
Find a box with a capacity of $1$ L. How many different ones can you make?

Composite solids

Split the solid into parts you know. For volume, add the parts (or subtract a hole). For surface area, be careful: faces that join two parts are hidden inside and don't count.

Example 4●●●

A grain silo is a cylinder of radius $3$ m and height $10$ m with a cone of height $2$ m on top. Find its volume, correct to two decimal places.

$V_1 = \pi \times 3^2 \times 10 = 90\pi$

Cylinder part.

$V_2 = \dfrac13 \pi \times 3^2 \times 2 = 6\pi$

Cone part, with the same radius as the cylinder.

$V = 96\pi \approx 301.59$ m³

Add, and only now use the calculator.

Your turn

A hemispherical salad bowl has an inside radius of $10$ cm. How many litres does it hold, to two decimal places?

Show the working

A hemisphere is half a sphere: $V = \dfrac12 \times \dfrac43 \pi \times 10^3 = \dfrac23 \pi \times 1000 \approx 2094.40$ cm³. Divide by $1000$: about $2.09$ L.

Flashcards

Volume of a pyramid or cone

$\dfrac13 \times$ base area $\times$ height

Surface area of a closed cylinder

$2\pi r^2 + 2\pi r h$

Volume of a sphere

$\dfrac43 \pi r^3$

$1000$ cm³

$1$ L

Practice

A box is $30$ cm by $20$ cm by $15$ cm. What is its capacity in litres?
Find the volume of a cylinder with radius $5$ cm and height $12$ cm, correct to two decimal places.
Find the surface area of the same closed cylinder, correct to two decimal places.
A square pyramid has a base of side $6$ cm and a height of $4$ cm. Find its volume.
For the same pyramid, find the total surface area.
Find the volume of a cone with radius $3$ m and height $4$ m, correct to two decimal places.
Find the volume of a sphere with radius $6$ cm, correct to two decimal places.
A pool holds $1$ m³ of water. In litres that is:
Exam style: A hemispherical bowl has an inside radius of $10$ cm. Find its capacity in litres, correct to two decimal places.
Exam style: A silo is a cylinder with radius $3$ m and height $10$ m, topped by a cone of height $2$ m. Find the volume of the silo in m³, correct to two decimal places.