# Area of a Frustum of a Cone Calculator

Created by Davide Borchia
Reviewed by Komal Rafay
Last updated: Aug 30, 2022

Subtracting cones generates an exciting shape: learn how to calculate the area of the frustum of a cone: this calculator will explain step-by-step how to do this!

• What is the frustum of a cone?
• The elements of the frustum of a cone: how to build one from two cones;
• How to calculate the surface area of the frustum of a cone;
• The formula for the surface area of the frustum of a cone in action: some basic examples.

## What is a frustum of a cone?

The frustum of a cone is a solid shape which remains after intersecting a cone with a plane parallel to the base. The shape of a frustum is pretty intuitive, and you can imagine it without much effort. The name, however, is not so well known.

Take a look at the drawing of a frustum of a cone.

This shape clearly comes from the subtraction of a smaller cone, with base radius $r$, from a larger one, with radius $R$. Here they are: A frustum of a cone and the two cones which generate it.

In the image above, you can clearly see the two cones. Their heights are:

• $H+h$, for the larger cone; and
• $h$ for the smaller cone.

To identify a frustum of a cone, we need to know a limited set of quantities:

• $R$ — The radius of the larger base;
• $r$ — The radius of the smaller base; and
• $H$ — The height of the frustum.

From these quantities, you can calculate the slant height of the frustum of a cone. In a "normal" cone, the slant height is the length of the segment connecting the apex and the circumference of the base. To find the slant height of a frustum of a cone, we need to subtract the slant heights of the two generating cones.

We calculate the slant height of a cone using the pythagorean theorem. For the smaller cone, we have:

$l = \sqrt{r^2 + h^2}$

For the larger cone, we find that:

$L = \sqrt{R^2 + (H+h)^2}$

The slant height of the frustum of a cone is then:

$L\!-\!l\! =\! \sqrt{\!R^2\!+ \!(H\!+\!h)^2\!} \!-\! \sqrt{\!r^2 \!+ \!h^2\!}$

We can call this quantity $S$.

🙋 Be careful: this equation is meaningful only if the values of $R$, $r$, $H$, and $h$ can define a frustum of a cone!

Alternatively, you can calculate the slant height of the frustum with the formula:

$S = L-l = \sqrt{H^2+(R-r)^2}$

## How to calculate the surface area of the frustum of a cone — formula for truncated cones

To find the total surface area of the frustum of a cone, we need to know how to calculate the area of a cone: the frustum's area will come right after!

To calculate the area of a cone with base radius $R$ and height $H$, we split the figure into two parts:

• The base; and
• The lateral surface.

We compute these areas separately and sum them to get the result:

\begin{align*} A_{\text{base}}& = \pi\!\cdot \!R^2\\ A_{\text{side}}& = \frac{1}{2}\!\cdot\! c\! \cdot\! L \\ &= \frac{1}{2}\! \cdot \!2\!\cdot\! R\!\cdot \!\pi\!\cdot \!L=R\!\cdot\!\pi\!\cdot\!L \end{align*}

Thus:

\begin{align*} A_{\text{cone}} &= A_{\text{base}}+A_{\text{side}} \\ & = \pi\cdot R^2 + \pi\cdot R\cdot L =\\ &=\pi\cdot R(R+L) \end{align*}

To calculate the surface area of the frustum of a cone, there are two choices:

• Calculate the surface area of the generating cones' elements, and build the elements of the frustum; or
• Subtract the surface areas of the generating cones, and apply appropriate corrections.

In our calculator for the area of the frustum of a cone, we implement the first one.

Which elements do you need if you want to calculate the total surface area of the frustum of a cone?

• The area of the frustum's larger base;
• The area of the frustum's smaller base; and
• The lateral surface area of the frustum of the cone.

Let's calculate the first two of these elements:

\begin{align*} A_{\text{base}_1} &= \pi\cdot R^2\\ A_{\text{base}_2} &= \pi\cdot r^2 \end{align*}

Well, that was super easy, barely an inconvenience. What about the lateral surface area of the frustum of the cone? We calculate it by subtracting the side surfaces of the generating cones:

\begin{align*} A_{\text{side}_{\text{F}}} &= A_{\text{side}_1} - A_{\text{side}_2} \\ & =R\!\cdot\!\pi\!\cdot\!L \!- \!r\!\cdot\!\pi\!\cdot\!l \end{align*}

The last step is rearranged a bit, considering that, to build a frustum, our quantities must satisfy the following condition: $R\cdot l = r \cdot R$. The equation for the side area of the frustum becomes:

\begin{align*} A_{\text{side}_{\text{F}}} &= R\!\cdot\!\pi\!\cdot\!L\!-\!R\!\cdot\!\pi\!\cdot\!l\\ &=R\!\cdot\!\pi\!\cdot\!L\!-\!R\!\cdot\!\pi\!\cdot\!l \!+\! R\!\cdot\!\pi\!\cdot\!l\\ &+ r\!\cdot\!\pi\!\cdot\!L\!-\!r\!\cdot\!\pi\!\cdot\!l \!-\!r\!\cdot\!\pi\!\cdot\!l\\ &=R\!\cdot\!\pi\!\cdot\!S\!+\!r\!\cdot\!\pi\!\cdot\!S\\ &+\red{\!R\!\cdot\!\pi\!\cdot\!l}-\red{\!r\!\cdot\!\pi\!\cdot\!L}\\ &=\pi\!\cdot\!S\!\cdot\!(R\!+\!r) \end{align*}

Notice that the last two elements cancel each other.

Now that we have collected all the elements: it's time to discover the formula for the surface area of the frustum of a cone:

\begin{align*} A&\!=\! A_{\text{base}_1}\!+\!A_{\text{base}_2}\!+\!A_{\text{side}_{\text{F}}}\\ &=\pi\!\cdot \!R^2\!+\!\pi\!\cdot \!r^2\!+\! (R\!+\!r)\!\cdot\!\pi\!\cdot\!S\\ &= \pi\cdot\!(R^2\!+\!r^2+\!R\!\cdot\!S) \end{align*}

## I-conic: a collection of tools about cones

You learned how to calculate the area of a frustum of a cone: but that's not all! You can also learn to calculate its volume and everything there is to know about cones and their derivate shapes.

Keep learning with:

## FAQ

### How do I calculate the area of the frustum of a cone?

To calculate the area of the frustum of a cone, separate the shape into the following three elements, with their respective areas:

1. The larger base: A₁ = π × R²;
2. The smaller base: A₂ = π × r²; and
3. The side of the frustum: As = π × S × (R + r);

The area of the frustum of a cone is:
A = A₁ + A₂ + As = π × (R² + r² + S × (R + r))

### What is the frustum area with radii 8, 2, and slant height 12?

The area is 591. To calculate it, use the formula for the area of the frustum of a cone:
A = π × (R² + r² + S × (R + r))
A = π × (8² + 2² + 12 ×10) = 591
You can also calculate the area of the frustum of a cone if you know other quantities: the height or the slant height of either the larger cone or, the smaller one.

### What do you mean by frustum of a cone?

A frustum (of any solid characterized by a base and an apex, so pyramids, cones, etcetera) is a solid obtained by cutting the original shape with two distinct parallel planes. This generates a pair of parallel bases connected by one — or more — slanted sides.

### Which shape do I obtain if I cut a cone with a tilted plane?

There is no definition for solids obtained by cutting a cone with non-parallel planes. The resulting base (if the plane doesn't intersect the base) would be either a circle (the shape is named frustum in this case) or an ellipse.

In case the plane interests the base, other conic sections would emerge: parabola and hyperbolas. There is no general formula for such shapes.

Davide Borchia
Frustum
Height (H)
in
in
in
Slant height (S)
in Area of the frustum
cm²
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