Year 11 General Shape And Measurement

Surface Area

20 practice questions 0 video lessons Theory + worked examples

Theory

Surface area is the total area of all the outside faces of a 3D solid. This page covers the surface area formulas for the cube, rectangular prism, cylinder, cone, sphere and hemisphere, with worked examples on painting costs and Pythagoras-based slant heights.

The surface area of a 3D solid is the total area of all of its outside faces, measured in square units (cm $^{2}$ , m $^{2}$ , etc.). One useful way to picture it: imagine peeling the surface off the solid and laying it flat — the surface area is the area of the resulting net.

For a prism or pyramid, surface area is the sum of the areas of each face. For curved solids like the cylinder, cone and sphere, there is a dedicated formula because the curved surface unrolls into a recognisable flat shape (a rectangle, a sector of a circle, and so on).

A hemisphere is half a sphere. Its surface area depends on whether you include the flat circular cut: the curved part alone is $2 π r^{2}$ , and including the flat base it becomes $3 π r^{2}$ . Always read the question carefully.

The net of a closed cylinder: two circles plus a rectangle.

A cone: the slant height

s

(not the perpendicular height

h

) is used in the surface area formula.

The standard surface area formulas:

Cube (side $s$ ) — six identical square faces:

S A = 6 s^{2}

SA = 6 s^{2}

Rectangular prism (length $ℓ$ , width $w$ , height $h$ ):

S A = 2 ℓ w + 2 ℓ h + 2 w h

SA = 2 ℓ w + 2 ℓ h + 2 w h

Closed cylinder (radius $r$ , height $h$ ) — two circular ends plus the curved side:

S A = 2 π r^{2} + 2 π r h

SA = 2 π r^{2} + 2 π r h

Cone (radius $r$ , slant height $s$ ) — base circle plus curved side:

S A = π r^{2} + π r s

SA = π r^{2} + π r s

Sphere (radius $r$ ):

S A = 4 π r^{2}

SA = 4 π r^{2}

Hemisphere. Curved part only: $2 π r^{2}$ . Curved part plus the flat circular base: $3 π r^{2}$ . Always check which version the question is asking for.

Cone with perpendicular height only. If the question gives the perpendicular height $h$ instead of the slant $s$ , find the slant first using Pythagoras: $s = \sqrt{r^{2} + h^{2}}$ .

How to find the surface area of any solid

Identify each face of the solid. For a prism or pyramid, list the flat faces. For a cylinder, cone or sphere, recognise the standard shape and use its formula.
Compute the area of each face using the matching formula (rectangle $= ℓ w$ , triangle $= \frac{1}{2} b h$ , circle $= π r^{2}$ , curved cylinder $= 2 π r h$ , curved cone $= π r s$ ).
Add all the face areas together, and write the answer in square units. If the solid is open, leave out the missing face(s).

Composite or unusual solids. Break the solid into recognisable pieces (e.g.\ a cylinder topped by a hemisphere), compute each piece's surface area, and add them — but subtract any face that is hidden where the pieces meet.

Example 1 — Rectangular Prism

A box is

8

cm long,

5

cm wide and

3

cm tall. Find its surface area.

Solution

Use the rectangular-prism formula.

$S A$	$=$	$2 ℓ w + 2 ℓ h + 2 w h$
$S A$	$=$	$2 (40) + 2 (24) + 2 (15)$
$S A$	$=$	$80 + 48 + 30$
$S A$	$=$	$158 {cm}^{2}$

SA = 158

Example 2 — Closed Cylinder

A closed cylinder has radius

4

cm and height

10

cm. Find its surface area to two decimal places.

Solution

$S A$	$=$	$2 π r^{2} + 2 π r h$
$S A$	$=$	$2 π (16) + 2 π (4) (10)$
$S A$	$=$	$32 π + 80 π$
$S A$	$=$	$112 π$
$S A$	$\approx$	$351.86 {cm}^{2}$

SA \approx 351.86

Example 3 — Cone (find slant first)

A cone has base radius

6

cm and perpendicular height

8

cm. Find its surface area to two decimal places.

Solution

Find the slant height by Pythagoras, then apply the cone formula.

$s$	$=$	$\sqrt{6^{2} + 8^{2}}$
$s$	$=$	$\sqrt{100} = 10$
$S A$	$=$	$π r^{2} + π r s$
$S A$	$=$	$36 π + 60 π$
$S A$	$=$	$96 π$
$S A$	$\approx$	$301.59 {cm}^{2}$

SA \approx 301.59

Example 4 — Painting Cost

The four walls and the ceiling of a room (

5

\times

4

\times

3

m) are to be painted. Paint covers

10

^{2}

per litre and costs

$ 22

per litre, sold in whole litres only. Find the cost.

Solution

Find the area to be painted, then convert to litres (rounding up).

walls	$=$	$2 (5) (3) + 2 (4) (3)$
walls	$=$	$30 + 24 = 54$
ceiling	$=$	$5 \times 4 = 20$
total	$=$	$74 m^{2}$
litres	$=$	$74 \div 10 = 7.4 \to 8$
cost	$=$	$8 \times $ 22 = $ 176$

cost = 176

Common pitfalls

Forgetting the $2 π$ on the curved side of a cylinder. The curved face has dimensions circumference

\times

height, which is

2 π r \times h

, not

r \times h

. Without the

2 π

the answer is way off.

Open vs closed cylinders. A "closed" cylinder has both ends. An "open" cylinder (a tube) has none. "Open at the top" (a paint can with no lid) has only one. Read the question word for word and add the right number of

π r^{2}

caps.

Perpendicular vs slant height on a cone. The cone surface area formula uses the slant height

s

. If only the perpendicular height

h

is given, find

s

using Pythagoras:

s^{2} = r^{2} + h^{2}

Mixed units. Convert lengths to the same unit before substituting. Mixing cm and m will give answers off by a factor of

100

10 000

Frequently asked questions

What is surface area?

Surface area is the total area of all the outside faces of a 3D solid, measured in square units like square centimetres or square metres. If you peeled the surface off the solid and laid it flat, the surface area would be the area of the resulting net.

What is the surface area formula for a cylinder?

For a closed cylinder with radius r and height h, the surface area is 2 pi r squared plus 2 pi r h. The two pi r squared is the area of the two circular ends, and 2 pi r h is the area of the curved side (its circumference times its height).

What is the difference between a closed cylinder and an open cylinder?

A closed cylinder has both circular ends, so its surface area is 2 pi r squared plus 2 pi r h. An open cylinder (like a tube) has no ends, so the surface area is just 2 pi r h. A cylinder open at the top (like a paint can with no lid) has only one end, so the surface area is pi r squared plus 2 pi r h.

Why does the cone formula use the slant height instead of the perpendicular height?

The curved surface of a cone unrolls into a sector of a circle whose radius is the slant height s, not the perpendicular height h. If you are only given the perpendicular height and the radius, find the slant height first using Pythagoras: s squared equals r squared plus h squared.

What is the surface area of a hemisphere?

It depends whether the flat circular face is included. The curved part alone is 2 pi r squared. Including the flat base, the total is 3 pi r squared. Always read the question to see which one is being asked for.

How do I work out how much paint to buy for a wall?

First calculate the total area to be painted in square metres. Divide by the coverage rate of the paint (square metres per litre) to get the number of litres needed. Then round up to the next whole litre because paint is sold in whole tins, and multiply by the cost per litre.