Surface Area of a Cone

In this tutorial, we'll learn how to find the surface area of a cone. But before we can get into the meat of it, let's get a few important things out of the way.

Here's what a cone typically looks like.

So a cone has a base that tapers smoothly into a point at the other end (called vertex).

Now the base can be something other than a circle as well. However, generally, when we say “cone”, we are referring to one with a circular base – a circular cone. And more specifically a right circular cone.

What is a Right Circular Cone?

A right circular cone is a cone with a circular base whose axis – the line joining the vertex to the base's center – is perpendicular to the base.

Note – In this tutorial, we are concerned only with the surface area of a right circular cone.

Surface Area of a Cone?

There are two surfaces that make up the total surface of the cone - the lateral (or curved) surface and the base. The surface area of the cone is the total area covered by these surfaces.

Alright, so what's formula for the surface area of a cone?

Formula for Surface Area of a Cone

For a cone with a radius $\hspace{0.2em} r \hspace{0.2em}$ , height $\hspace{0.2em} h \hspace{0.2em}$ , and slant height $\hspace{0.2em} l \hspace{0.2em}$ ,

1. Total surface area

\text{TSA} = \pi r (l + r)

2. Lateral surface area or curved surface area

\text{LSA} = \pi rl

3. Base area

A_{base} = \pi r^2

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Derivation

If you cut open the cone and separate its curved surface and base, you will end up with a sector of a circle and a complete circle, as shown below.

So the area of the sector would give us the curved or lateral surface area (LSA) of the cone. Right?

Now, the radius of the sector would be the same as the slant height of the cone. And the arc length of the sector would be the same as the circumference of the cone's base.

Using the formula for the area of a sector, we can get the curved or lateral surface area (LSA) of the cone.

\begin{align*} \text{LSA} &= A_{sector} \\[1.3em] &= \frac{\text{radius} \cdot \text{arc length}}{2} \\[1.3em] &= \frac{l \cdot 2 \pi r}{2} \\[1.3em] &= \pi r l \end{align*}

Next, the base is a circle, so its area is straightforward -

A_b = \pi r^2

Finally, adding the two from above we get the total surface area (TSA).

\begin{align*} \text{TSA} &= \text{LSA} + A_{b} \\[1em] &= \pi r l + \pi r^2 \\[1em] &= \pi r (l + r) \end{align*}

How to Find the Surface Area of a Cone

In this section, we'll use the formulas from above to solve example problems based on the surface area of a cone.

Example

A cone has a base radius of $\hspace{0.2em} 5 \hspace{0.2em}$ cm and a slant height of $\hspace{0.2em} 8 \hspace{0.2em}$ cm. Find its curved surface area and also its total surface area.

Solution

Plugging the values of the base radius ( $\hspace{0.2em} r = 5 \hspace{0.2em}$ ) and slant height ( $\hspace{0.2em} l = 8 \hspace{0.2em}$ ) in the formula for the curved or lateral surface area of a cone, we have

\begin{align*} \text{LSA} &= \pi r l \\[1em] &= \pi \cdot 5 \cdot 8 \\[1em] &\approx 125.66 \end{align*}

Similarly, for the total surface area -

\begin{align*} \text{TSA} &= \pi r (l + r) \\[1em] &= \pi \cdot 5 \cdot (8 + 5) \\[1em] &\approx 204.2 \end{align*}

So the curved and total surface areas of the cone are $\hspace{0.2em} 125.66 \text{ cm}^2$ and $\hspace{0.2em} 204.2 \text{ cm}^2$ .

Example

Find the lateral surface area of a cone with a base radius of $\hspace{0.2em} 3 \hspace{0.2em}$ and a height of $\hspace{0.2em} 4 \hspace{0.2em}$ .

Solution

The formula for the lateral surface area involves the radius and the slant height.

\text{LSA} = \pi r l

But while the question gives us the radius ( $\hspace{0.2em} r = 3 \hspace{0.2em}$ ) and height ( $\hspace{0.2em} h = 4 \hspace{0.2em}$ , it doesn’t give us the slant height $\hspace{0.2em} l$ . So, we need to calculate it.

As you can see in the figure above, the radius, height, and slant height form a right triangle. So applying the Pythagorean theorem, we have –

\begin{align*} l &= \sqrt{r^2 + h^2} \\[1em] &= \sqrt{3^2 + 4^2} = 5 \end{align*}

And now we can plug $\hspace{0.2em} r \hspace{0.2em}$ and $\hspace{0.2em} l \hspace{0.2em}$ into our formula.

\begin{align*} \text{LSA} &= \pi \cdot 3 \cdot 5 \\[1em] &= 47.12 \end{align*}

And with that, we come to the end of this tutorial on the surface area of a cone. Until next time.