Surface Area

What is surface area?

A face is one of the flat or curved surfaces that make up a 3D shape
The surface area of a 3D shape is the sum of the areas of all the faces that make up the shape
Note how we are carrying a 2D idea (area) into 3 dimensions here

How do I find the surface area of cuboids, pyramids, and prisms?

In cuboids, polygonal-based pyramids, and polygonal-based prisms (ie. pyramids and prisms whose bases have straight sides), all the faces are flat
The surface area is found simply by adding up the areas of these flat faces
When calculating surface area, it can be very helpful to draw a 2D net for the 3D shape in question
For example:
- The base of a square-based pyramid is 15 cm on a side
- The triangular faces are identical isosceles triangles, each with a height (from the base to the top of the pyramid) of 23 cm
- Find the total surface area of the pyramid
- Draw a net for the shape:

Net-Square-Based-Pyramid, IGCSE & GCSE Maths revision notes

- Area of square base =15² = 225 cm²
- Area of one triangular face = ½ base × height = ½ × 15 × 23 =172.5 cm²
- Total surface area =225 + 4 × 172.5 = 915 cm²

How do I find the surface area of cylinders, cones, and spheres?

All three of these shapes have curved faces, so we have to be a little more careful when calculating their surface areas

Surface area of a cylinder formula

The net of a cylinder consists of two circles and a rectangle:
The total surface area of a cylinder with base radius r and height h is therefore given by the formula:
Total surface area of a cylinder = 2πr² + 2πrh

Surface area of a cone formula

The net of a cone consists of the circular base along with the curved surface area:
The length $l$ in this diagram is known as the slant height (while h is the vertical height of the cone)
To find the surface area of a cone with base radius r and slant height l, we use the formulae:
- Curved surface area of a cone = πrl
- Total surface area of a cone = πr²+πrl

Surface area of a sphere formula

To find the surface area of a sphere with radius r, use the formula:
Surface area of a sphere=4πr²

Sphere Radius r, IGCSE & GCSE Maths revision notes

Examiner Tip

The formula for the surface area of a sphere or the curved surface area of a cone will be given to you in an exam question if you need it
The rest of the formulae here come from what you should already know about areas of rectangles, triangles, and circles

Be careful when calculating the surface area of a hemisphere:The total surface area consists of the curved part (half of a sphere) PLUS the flat circular face – so the total surface area is 3πr²

Worked example

The base radius, $r$ , of a cone is the same as the radius of a hemisphere. The total surface area of the cone is equal to the total surface area of the hemisphere.

(a)

Find the slant height,

l

, of the cone in terms of

r

Find an expressions for the surface area of the hemisphere in terms of

l

and

r

.
Remember that a hemisphere has both a curved surface area and a flat circular face so the formula for the surface area is:

Surface area of hemisphere =

\frac{1}{2} \times 4 π r^{2} + π r^{2} = 3 π r^{2}

Find an expressions for the surface area of the cone in terms of

l

and

r

.
Remember that a cone has both a curved surface area and a flat circular face so the formula for the surface area is:

Surface area of cone =

π r l + π r^{2} = π r (l + r)

The surface areas are equal, so set these two formulae equal to each other.

3 π r^{2} = π r (l + r)

Rearrange to make

l

the subject.

Begin by dividing both sides by

π r

\begin{array}{rcl} 3 {π r}^{2} & = & π r (l + r) \\ 3 r & = & l + r \end{array}

l = 2 r

(b)

Given that

r = 19

cm, find the curved surface area of the cone.
GIve your answer accurate to 1 decimal place.

Use your answer from part (a) to find the value of

l

, by substituting

r = 19

into

l = 2 r .

l = 2 r = 2 \times 19 = 38 cm

Substitute

r = 19

and

l = 38

into the formula for the curved surface area of the cone.

Note that this is not for the whole surface area.

π r l = π \times 19 \times 38 = 722 π = 2268.229 . . . .

Round your answer to 1 decimal place.
The first decimal place is a 2, and this is followed by a 2 so you do not need to round it up.

Curved surface area = 2268.2 {cm}^{2} (1 d . p .)

Surface Area (CIE IGCSE Maths: Extended)

Revision Note

Surface Area

What is surface area?

How do I find the surface area of cuboids, pyramids, and prisms?

How do I find the surface area of cylinders, cones, and spheres?

Surface area of a cylinder formula

Surface area of a cone formula

Surface area of a sphere formula

Examiner Tip

Worked example

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Author: Jamie W