Volume and surface area of a cone

A cone has a circle at the bottom and a point at the top. To find its volume we need the height and the radius of the circle at the bottom. To find its surface area we also need the length of the side of the cone.

When do I use this?

Whenever a task asks how much fits inside a cone shape, or how much material covers one. The difference between the two, inside and outside, is explained in 3D shapes: volume and surface area. In the GCSE exam, both cone formulas are given to you in the question when you need them, so the job is to know what the letters mean.

The letters on a cone

Three lengths matter:

• $r$, the radius of the circle at the bottom.
• $h$, the height, straight up from the middle of the circle to the point.
• $l$, the length of the side of the cone, running along the outside from the edge of the circle up to the point.

The height and the side length are two different lengths. The side is the longer one, because it slopes.

The volume of a cone

To find the volume of a cone, we need the height and the radius of the circle at the bottom. We use this formula:

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

Take a cone with radius $3$ cm and height $4$ cm. Then

$$V = \frac{1}{3} \times 4 \times \pi \times 3^2 = \frac{1}{3} \times 4 \times \pi \times 9 = 12\pi \approx 37.70 \text{ cm}^3$$

So about $37.70$ cubic centimetres fit inside the cone. If the question wants the answer in terms of $\pi$, it is $12\pi$ cm³.

The surface area of a cone

The surface area can be found if we also have the length of the side of the cone. Then we use this formula:

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

The first part, $\pi \times r \times l$, is the curved surface of the cone. The second part, $\pi \times r^2$, is the circle at the bottom.

For the same cone, with $r = 3$ cm and a side length of $l = 5$ cm:

$$S = \pi \times 3 \times 5 + \pi \times 3^2 = 15\pi + 9\pi = 24\pi \approx 75.40 \text{ cm}^2$$

So about $75.40$ square centimetres would cover the cone, bottom included.

Common mistakes

• Using the height where the formula wants the side length. The volume uses the height $h$. The surface area uses the length of the side, $l$. They are different lengths on the cone, so check which one you have been given.
• Mixing up volume and surface area. Volume is what fits inside, in cm³. Surface area is what is around the outside, in cm².
• Giving the answer in the wrong unit. If a volume comes out in cm², or a surface area in cm³, we have probably calculated wrong somewhere.

Related

The full catalogue of shapes is in 3D shapes: volume and surface area. The cylinder with the same circle at the bottom is in volume and surface area of a cylinder. The pyramid also divides by 3, in volume of a pyramid. The circle at the bottom is explained in how to find the area of a circle.