3D shapes: volume and surface area

The volume of a 3D shape is how much fits inside it. That could be how much fits inside a box, a cylinder, a pyramid or a similar shape.

We have a whole set of formulas for how much fits inside the different shapes, but why the formulas look the way they do can be quite advanced. So we'll stick to the formulas, and not how we got to them. You only start looking at where the formulas come from later, at A level.

What surface area means

The surface area of a shape is the area of all its sides. So, how much there is around the outside of the shape. A good way to think about it is this: if we had to wrap the shape in wrapping paper, how much wrapping paper would we need?

That is the difference in one sentence. Volume is what is inside the shape, surface area is what is around it. They are two different things, and they have two different formulas for every shape.

The shapes and their formulas

Every shape has a small set of lengths you need to know, a height, a length, a radius. Those are the letters that go into its formulas, and nothing else. Here is the catalogue. Each shape has its own guide with a worked example.

The cuboid

The cuboid is the box. Its volume is the height, the length and the width multiplied together:

$$V = h \times l \times w$$

We can see it as finding the area of the bottom of the box and then multiplying by how tall the box is. The surface area is

$$S = 2 \times (w \times h + l \times w + l \times h)$$

where we're really just adding up the area of all the sides. See volume and surface area of a cuboid.

The cylinder

The volume of a cylinder can be found if we have the height and the radius:

$$V = \pi \times r^2 \times h$$

What we're actually doing is finding the area of the circle at the bottom and then multiplying by how tall the cylinder is. The surface area is the curved surface, the tube, plus the circle at the bottom and the circle at the top:

$$S = 2 \times \pi \times r \times h + 2 \times \pi \times r^2$$

See volume and surface area of a cylinder.

The cone

To find the volume of a cone we need the height and the radius of the circle at the bottom:

$$V = \frac{1}{3} \times h \times \pi \times r^2$$

The surface area also needs the length of the side of the cone, $l$:

$$S = \pi \times r \times l + \pi \times r^2$$

See volume and surface area of a cone.

The pyramid

The volume of a pyramid can be found if we can work out the area of the base and we have the height of the pyramid. Remember that the pyramid can have different shapes as its base: a triangle, a quadrilateral or a pentagon. We multiply the base by the height and then divide by 3:

$$V = \frac{A_{\text{base}} \times h}{3}$$

See volume of a pyramid.

The sphere

We can find the volume of a sphere if we just have the radius:

$$V = \frac{4}{3} \times \pi \times r^3$$

and the surface area with

$$S = 4 \pi \times r^2$$

See volume and surface area of a sphere.

The frustums

A frustum is a cone or a pyramid with the top cut off. Both have a volume formula, which you'll find in the guide volume of a frustum. Their surface areas are quite involved, and in most tasks you won't meet them.

Volume is cubic, surface area is square

Area comes in square units, like square metres, because we multiply two lengths. Volume comes in cubic units, like cubic metres, because we multiply three lengths. So a volume always ends in cm³ or m³, and a surface area in cm² or m².

That is worth keeping an eye on. It is important that we get the answer in the right unit, otherwise we've probably calculated wrong somewhere. The glossary on units of measurement goes into square metres and cubic metres.

A short word about the formulas

Many of these formulas for volume and surface area can look a bit difficult. Many of them you will probably never use either. Later on, at A level, you look at some of these formulas again and prove why they are true and can be used. In most tasks you won't meet the surface area of a frustum, because they are quite hard.

For GCSE there is one more reassurance. The formulas for the volume and surface area of a sphere and a cone are given to you in the question when you need them, and the general prism formula, area of the cross-section times the length, is on the formula sheet. So the job is to know what the letters mean and to put the right numbers in the right places.

Common misunderstandings

• "Volume and surface area are the same thing." No. Volume is how much fits inside the shape. Surface area is the area of all the sides, how much is around the outside, the wrapping paper. Every shape has a separate formula for each.
• "I have to remember all of these formulas for the test." You don't. Many of them you'll never use, and at GCSE the sphere and cone formulas are given to you. What you do need is to know which length is which on the shape.
• "The unit doesn't matter as long as the number is right." The unit is the check. A volume in cm² or a surface area in cm³ means something went wrong in the calculation.

Related guides

• Volume and surface area of a cuboid
• Volume and surface area of a cylinder
• Volume and surface area of a cone
• Volume of a pyramid
• Volume and surface area of a sphere
• Volume of a frustum
• The circle formulas that the cylinder and cone build on are in circles.