**The Volume of Cylinders**

Learn how to find the volume of a cylinder with this teaching wiki. Also, find links to plenty of useful volume of cylinder resources.

There are a few things to remember when understanding how to work out the volume of cylinders. First, a cylinder is a circular prism. Like other prisms, it is a two-dimensional shape stretched into a third dimension.

Both ends and any cross-section (a cross-section is what you get if you cut the cylinder at any point along its length) are identically sized circles.

It means you can calculate the volume of a cylinder (the space contained within the 3D shape) like any other prism: find the area of the end, then multiply that by the third dimension (the height or the depth, depending on the orientation).

Example 1:

**Find the volume of the cylinder below. Give your answer to 1 decimal place.**

We’ll start by finding the area of the circular face. It has a radius of 4m, so we can substitute that into the formula for the area of a circle:

- Area = π × r²
- Area = π × 4²
- Area = 50.265… m²

The question asks for an answer to 1 decimal place, but we still have another calculation, so we won’t round yet.

We can imagine our cylinder as a circle of area of about 50m2 that has been stretched downwards by 10m. To find the volume, we take our area and multiply it by the height:

- The volume of a prism = area of cross-section × height
- Volume = 50.265… × 10
- Volume = 502.65…
- Volume = 502.3 m³ (to 1 decimal place)

There are two things to note about this answer. Firstly, if we’d rounded our area to 1 decimal place (50.3), then we would have got a volume of 503m3 instead of 502.3m3 – in an exam, you’d lose a mark for that.

Secondly, our units. Any volume must be given in cubed units, for example, m3, cm3, or m3, or sometimes in liters (as 1 liter is the same as 1000 cm3).

Example 2:

**A sweet, cylindrical container has a length of 12cm and a diameter of 1.5cm. Find its volume. Give your answer to 1 decimal place.**

This time you need to be careful. We need the radius to find a circle’s area or a cylinder’s volume. In this case, we have been given the diameter. The radius is half the diameter:

- radius = diameter ÷ 2
- radius = 1.5 ÷ 2
- radius = 0.75cm

This time, we will use a formula for the volume. The volume of a cylinder is the area of the front face multiplied by the height (h) or length (l). Therefore we can say:

- Volume = Area × length
- Volume = π × r² × l
- Volume = πr²l

Substituting our values gives us the following:

- Volume = π × 0.75² × 12
- Volume = 21.2cm³ (1d.p.)

Example 3:

**A cylindrical grain silo has a radius of 10m and a height of 15m. Find its volume in terms of π.**

Because π is an irrational number (it never ends and never repeats), you will have to round your answer any time you calculate using it. Sometimes, you might need to use your solution in future analyses, so you don’t want to round it. In this case, you can leave your answer in terms of π. It is also how questions are often presented in non-calculator exams.

First, we can find the area of the top of our cylinder, as usual:

- Area = π × r²
- Area = π × 100
- Area = 100π m²

When leaving our answer in terms of pi, we do the other calculations (102 = 100) without multiplying by pi. Pi is still in our response, and if we wanted, we could use it to calculate the volume numerically.

We have the area of the top (and bottom) of our cylinder; now we multiply by the height:

- The volume of a prism = area of a cross-section × height
- Volume = 100π × 15
- Volume = 1500π m³

At this point, we have our answer in terms of pi. Therefore, we don’t need to do any further calculations.