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.

A cone

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.

Right and Oblique Cones

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.

Cone - Dimensions

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 r\hspace{0.2em} r \hspace{0.2em}, height h\hspace{0.2em} h \hspace{0.2em}, and slant height l\hspace{0.2em} l \hspace{0.2em},

1. Total surface area

TSA=πr(l+r)\text{TSA} = \pi r (l + r)

2. Lateral surface area or curved surface area

LSA=πrl\text{LSA} = \pi rl

3. Base area

Abase=πr2A_{base} = \pi r^2
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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.

Surface Area of a Cone - Derivation

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.

LSA=Asector=radiusarc length2=l2πr2=πrl\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 -

Ab=πr2A_b = \pi r^2

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

TSA=LSA+Ab=πrl+πr2=πr(l+r)\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.


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


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

LSA=πrl=π58125.66\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 -

TSA=πr(l+r)=π5(8+5)204.2\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 125.66 cm2\hspace{0.2em} 125.66 \text{ cm}^2 and 204.2 cm2\hspace{0.2em} 204.2 \text{ cm}^2.


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


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

LSA=πrl\text{LSA} = \pi r l

But while the question gives us the radius (r=3\hspace{0.2em} r = 3 \hspace{0.2em}) and height (h=4\hspace{0.2em} h = 4 \hspace{0.2em}, it doesn’t give us the slant height l\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 –

l=r2+h2=32+42=5\begin{align*} l &= \sqrt{r^2 + h^2} \\[1em] &= \sqrt{3^2 + 4^2} = 5 \end{align*}

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

LSA=π35=47.12\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.