IMAT Mathematics · Geometry

Solids: Surface Area, Volume and Scaling

Volumes and surface areas of prisms, cylinders, cones, spheres and pyramids, and how they change when every length is scaled.

All Geometry notes

Why this matters

The 2024 paper asked for the volume of a cylinder from its radius and height, and a 2014 question added up the surface areas of spheres of different radii.

Concept 1 of 2: Volume and surface area of the standard solids

A prism or cylinder has the same cross-section all the way along, so its volume is that area times the length. A cone or pyramid tapers to a point and holds exactly one third of the matching prism or cylinder. Surface area is the area of the net: every face laid out flat.

Definition

The height $h$ of a cone or pyramid is measured straight up from the base to the apex. The slant height $l$ of a cone runs along its side, and $l^2 = r^2 + h^2$ .
Total surface area includes the base(s); curved surface area is the curved part only.
Volume units are cubed ( $\text{cm}^3$ ); area units are squared ( $\text{cm}^2$ ). $1\ \text{litre} = 1000\ \text{cm}^3$ .

Solid	Volume	Surface area
Prism, cross-section area $A$ , length $L$	$AL$	Two end faces plus the rectangles round the side
Cuboid $a \times b \times c$	$abc$	$2(ab + bc + ca)$
Cylinder, radius $r$ , height $h$	$\pi r^2 h$	$2\pi r^2 + 2\pi r h$ (curved part $2\pi r h$ )
Cone, radius $r$ , height $h$ , slant $l$	$\tfrac{1}{3}\pi r^2 h$	$\pi r^2 + \pi r l$ (curved part $\pi r l$ )
Sphere, radius $r$	$\tfrac{4}{3}\pi r^3$	$4\pi r^2$
Pyramid, base area $B$ , height $h$	$\tfrac{1}{3}Bh$	Base plus the triangular faces

Practice this conceptself-check · 4 quick reps

A cone's volume uses the vertical height, not the slant height

V = \tfrac{1}{3}\pi r^2 h

needs the height measured straight up. If you are given the slant height, find

h

with Pythagoras first. The slant height belongs only in the curved surface area

\pi r l

Sphere: 4πr² for area, (4/3)πr³ for volume

The surface area of a sphere is

4\pi r^2

, with no

\tfrac{1}{3}

. Mixing the two formulas gives a wrong answer with the wrong units: area must come out in squared units, volume in cubed units.

Concept 2 of 2: How area and volume scale with length

Double every length of a cube and you can fit 4 small faces on each big face and 8 small cubes inside the big one. In general, scaling lengths by

k

scales areas by

k^2

and volumes by

k^3

. The mass of a solid object goes with its volume.

Definition

If two shapes are similar with length scale factor $k$ :

Every length (edges, radii, heights, perimeters) is multiplied by $k$ .
Every area (surface area, cross-section) is multiplied by $k^2$ .
Every volume (and the mass, for the same material) is multiplied by $k^3$ .
This holds only when all lengths scale. Changing only the radius of a cylinder at fixed height multiplies its volume by $k^2$ , not $k^3$ .

Scale factors

\frac{A_2}{A_1} = k^2 \qquad \frac{V_2}{V_1} = k^3

$k$ ratio of matching lengths, large to small

Worked example

Two similar jugs have heights 12 cm and 18 cm. The smaller holds 0.8 litres and needs

150\ \text{cm}^2

of glaze. How much does the larger hold, and how much glaze does it need?

Practice this conceptself-check · 4 quick reps

Volume goes with the cube of the length

Doubling every length makes a solid 8 times as heavy, not 2 or 4 times. A percentage change in length must be cubed for volume and squared for area, never just multiplied by 3 or 2.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

How area and volume scale with length
Scale factors
$\frac{A_2}{A_1} = k^2 \qquad \frac{V_2}{V_1} = k^3$

Reference tables (1)

Volume and surface area of the standard solids6 rows

Solid	Volume	Surface area
Prism, cross-section area $A$ , length $L$	$AL$	Two end faces plus the rectangles round the side
Cuboid $a \times b \times c$	$abc$	$2(ab + bc + ca)$
Cylinder, radius $r$ , height $h$	$\pi r^2 h$	$2\pi r^2 + 2\pi r h$ (curved part $2\pi r h$ )
Cone, radius $r$ , height $h$ , slant $l$	$\tfrac{1}{3}\pi r^2 h$	$\pi r^2 + \pi r l$ (curved part $\pi r l$ )
Sphere, radius $r$	$\tfrac{4}{3}\pi r^3$	$4\pi r^2$
Pyramid, base area $B$ , height $h$	$\tfrac{1}{3}Bh$	Base plus the triangular faces

Watch out for (3)

A cone's volume uses the vertical height, not the slant height
→ Volume and surface area of the standard solids
Sphere: 4πr² for area, (4/3)πr³ for volume
→ Volume and surface area of the standard solids
Volume goes with the cube of the length
→ How area and volume scale with length

Drill the past-year questions on this topic

Open the bank with this topic pre-filtered.

Open in the bank