## What you need to know

The **surface area** of a 3D shape is a measure of how much area the surfaces of that shape have in total. You can sometimes imagine unfolding the shape and flattening it out in order to measure the area of the resulting 2D shape, but this is not always possible. Shapes that you have to work out the surface area of can fall into one of two categories.

– All the faces are **flat** – in this case, working out the surface area of the shape amounts to working out the surface area of each of the flat faces (rectangles, triangles, etc) using the formulae you know, and then adding the results together.

– Some/all of the faces are **curved** – in this case, we will work out any flat face areas as normal, but then there will be some extra rule/formula to work out the area of the curved face, and we will go through those possibilities here.

A typical example of the first option is a pyramid.

**Example: **Below is a square-based pyramid. All 4 of its triangular faces are identical. Each triangular face has base 5cm and height 8cm. Work out the surface area of the shape.

Since each 4 triangular faces are identical, we only need to work out the area of one and then multiply it by 4 to get the area of them all.

The area of a triangle is \frac{1}{2}bh, and here the base is 5cm and the height is 8cm, so the area of one triangle is

\dfrac{1}{2}\times5\times8=20\text{ cm}^2

Therefore, the area of all 4 triangles is

4\times20=80\text{ cm}^2.

That almost covers the whole shape. All that’s left is the under-side – the base. We are told that it is a square, and we can see that the square has side-length 5cm, therefore its area is

5^2=25\text{ cm}^2

Therefore,

\text{total surface area }=25+80=105\text{ cm}^2.

The next examples fall into category 2 – they have curved faces. First up is a **cylinder**. This is a method you do have to memorise.

**Example: **Below is a cylinder. The circular face has radius 12mm and length of the cylinder is 25mm. Work out the surface area of the shape to the nearest whole number.

Cylinders have 2 flat, circular faces, and 1 curved face. The formula for the area of a circle is \pi r^2, and since the diameter of these circles is 12mm, the radius must be 6mm. Therefore, the area of 1 circle is

\pi\times6^2=36\pi

Thus, the area of the two circles is 2\times36\pi=72\pi (we’ll deal with the rounding at the end of the question).

Fortunately, the area of the curved face is sneakily easy. It’s actually just a folded-up rectangle. To understand this, take a piece of paper and roll it up – it forms a cylinder with the two ends missing. So, what we’re going to do is imagine rolling out a cylinder into a flat piece of paper.

If you actually do this yourself, you’ll see that firstly, the length of the cylinder is the length of the rectangle. Secondly (maybe surprisingly) the **circumference** of the circular face of the cylinder is the width of the rectangle. So, the area is just the circumference times by the length.

The circumference of a circle is 2\pi r, and r=6, so we get

\text{area of curved face}=(2\times\pi\times6)\times25=300\pi

Therefore, adding the areas of the 3 surfaces together, we get

\text{surface area }=72\pi+300\pi=1,169\text{ mm}^2 \text{ (nearest whole number)}

The last two shapes we’ll look at are a **cone **and a **sphere**. If a question on the surface area of these shapes comes up you will be given the formulae, but you should be comfortable with using them regardless (so do the questions given below!).

**Example: **Below is a cone. The radius of the base of the cone is 40cm, and the slant height of the cone is 55cm. Work out the surface area of the cone to 1dp.

The formula for the surface area of a cone is

\text{surface area of a cone }=\pi r^2+\pi rl

Where r is the radius of the base and l is the slant height. Therefore, the surface area of this cone is

\pi\times 40^2+\pi\times40\times55=11,938.1\text{ m}^2\text{ (1dp)}

**Example: **Below is a sphere with surface area 1,025\text{m}^2. Work out the length of the radius to 2sf.

The **formula for the surface area of a sphere **is

\text{surface area of a sphere }=4\pi r^2

We know that the surface area is 1,025, so we can set the formula above equal to 1,025 to get

4\pi r^2=1,025

Then, if we divide both sides by 4\pi, we get

r^2=1,025\div4\pi=81.5669...

Finally, square rooting this we get

r=\sqrt{81.5669...}=9.0\text{ cm}^2 \text{ (2sf)}.

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### Example Questions

1) Below is a cuboid with length 6mm, width 2.5mm, and height 4mm. Work out the total surface.

A cuboid has 6 flat, rectangular faces, and we will need the areas of all of them. Specifically, there are 3 pairs of faces since the front and back faces are the same, the top and bottom faces are the same, and the left and right faces are the same.

Firstly, the front face is a rectangle with height 4mm and width 2.5mm, so it has area

4\times2.5=10\text{ mm}^2

Therefore, the back face also has area 10\text{ mm}^2.

The face on the right has length 6mm and height 4mm, so it has area

6\times4=24\text{ mm}^2

Therefore, the left face also has area 24\text{ mm}^2

The face on the top has length 6mm and width 2.5mm, so it has area

6\times2.5=15\text{ mm}^2

Therefore, the bottom face also has area 15\text{ mm}^2.

Thus, adding all these values together, we get the total surface area to be

10+10+24+24+15+15=98\text{ mm}^2

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2) Below is a cone with surface area 120\text{cm}^2. Its base has radius 3cm. Work out the slant height of the cone to 1dp.

The formula for the **curved** surface area* of a cone is \pi rl, where r is the radius of the base and l is the slant height.

*this is not the whole surface area, just the curved section.

We know the whole surface area is 120\text{cm}^2 and we also know the radius. To work out the slant height, we need to first work out what the curved surface area is. In other words, we need to subtract the surface area of the base of the cone (since that’s the only other face) from 120 to get the curved surface area. The base is a circle, so its area is

\pi \times 3^2=9\pi\text{ cm}^2

Subtracting this from the total we have

120-9\pi=91.725...

This is written as a decimal to give an idea of how big it is, but when you put it into your calculator you should either use the ANS key to store the value, or you should directly type in 120-9\pi. If you don’t, it could affect your final answer.

Now, this must the area of the curved face, and the formula for the area of the curved face is given to us: \pi rl = 3\pi l (since we know r=3). So, setting this formula equal to the value we worked out, we get

3\pi l=120-9\pi

Then, to find the slant height, we will divide both sides by 3\pi to get

\dfrac{120-9\pi}{3\pi}=9.7\text{ cm (1dp)}.

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3) Josh is painting a very large spherical sculpture. The sculpture has radius 8.5m. He is using a special type of paint that costs £9.60 per pot, and each pot covers exactly **ten square metres**. How much will he have to spend in order to paint the whole sphere?

The formula for the surface area of a sphere is 4\pi r^2 where r is the radius.

We need to firstly work out the surface area of the sphere. So, with r=8.5, we get

\text{surface area }=4\times\pi\times(8.5)^2=907.920...\text{ m}^2

Now, since each pot covers ten square metres, we must divide this value by 10 to see how many pots he will require:

907.920... \div 10=90.720...

Since he can’t buy exactly 90.720… pots, he will have to buy 91 to cover the whole sphere. Therefore, the cost of all his paint will be

91\times 9.60=\pounds873.60

That’s a lot of paint.

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