Surface Area of Cone

The surface area of a cone is the total surface area covered by both of its sides, so the sum of the area of its circular base and its curved surface.

You should try imagining what a cone looks like, think of the body or the sides of a cone. This would give you an idea of the task.

Curved surface area of a cone

The curved surface area of a cone is the area of the cone's body without the base. Here the slant height of the cone is very important.

Illustrating the curved surface area of a cone, StudySmarter Originals

Calculating the curved surface area of a cone

The curved surface area of a cone is calculated by multiplying pi, the radius and slant height of a cone.

Surface area of a cone formula

As stated before, the surface area of a cone is the total combined surface area of its curved surface and circular base, so we can make some logical assumptions as to what the formula might be, but we'll go into the derivation of the formula soon. Here, however, is the formula you must know:

$a={\mathrm{\pi r}}^2+\mathrm{\pi rl}$

In this case, "a" is the total surface area, "r" is the radius of the circular base and "l" is the length of the curved surface (usually called slant height). l is not the internal height, they are two different measurements. The image below shows this in the case of a cone, to give you a better understanding.

A labelled diagram of a cone, StudySmarter Originals

If you are given the internal height of a cone, you can use the Pythagorean theorem to calculate the slant length.

An illustration on how the slant height is derived from the radius and the height, StudySmarter Originals

Surface area of cone derivation

Now that we know the formula, we should talk about how we can derive it from some other bits of information. Assuming we split the side (slant height side) of a cone and spread it out, we have what is displayed in the diagram below.

The main thing we need to remember is that a cone can be broken down into two sections, the circular base and the conical section or curved surface.

An illustration on the derivation of the total surface area of a cone, StudySmarter Originals

1. Separate the curved surface and the circular base. It's easier for you to calculate the surface area of each part separately. Forget about the circle section, for now, you will come back to it.
2. If you take the conical section and unfold it, you will see that it is actually a sector of a larger circle that has a radius of l. The circumference of this larger circle is therefore$2\mathrm{\pi l}$and the area is${\mathrm{\pi l}}^2$. The length of the arc of the sector you have is the same length as the circumference of the original circle section, which is$2\mathrm{\pi r}$.

3. The ratio between the area of the whole circle and the ratio of the area of the sector is the same as the ratio between the entire circumference and the part of the circumference of the sector. If you take the area of the sector to be "a", you can put this into an equation: \[\frac{a}{whole\, circle\, area}=\frac{arc\, length}{whole\, circle\, circumference}\]

4. We substitute the values from step 2 into the word equation from step 3: $\frac{a}{{\mathrm{\pi l}}^2}=\frac{2\mathrm{\pi r}}{2\mathrm{\pi l}}$

5. In this step, we're just going to look at what we need to do to simplify the above equation.

   The$2\mathrm{\pi }$ on the right-hand side both cancel:

   $\frac{a}{{\mathrm{\pi l}}^2}=\frac{\cancel{2\mathrm{\pi }}\mathrm{r}}{\cancel{2\mathrm{\pi }}\mathrm{l}}$

   Then we multiply both sides by ${\mathrm{\pi l}}^2$:

   $a=\frac{r}{l}{\mathrm{\pi l}}^2$

   This allows us to cancel out some l's:

   $a=\frac{r}{\cancel{l}}{\mathrm{\pi l}}^{\cancel{2}}$

   And that leaves us with:

   $a=\mathrm{\pi rl}$

6. Remember our circle from earlier? Well, the area of a circle is ${\mathrm{\pi r}}^2$ and the area of our conical section is $\mathrm{\pi rl}$, so if we take both of these areas and combine them we get the total surface area of a cone, which is:

Finding the surface area of a cone

Examples of surface of cones

In order to improve your ability in solving questions on surface of cones, you are advised to practice more problems.