# Area of a Frustum of a Cone Calculator

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!

In this article, you will learn everything about this interesting shape:

**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:

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 find the slant height of a cone by identifying the elements of a right triangle, and calculating the pythagorean theorem. For the smaller cone, we have:

For the larger cone, we find that:

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

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:

## 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:

Thus:

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:

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:

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:

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:

## 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:

- The
**cone volume calculator**; - The
**truncated cone volume calculator**; and - The
**truncated cone calculator**;

## 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:

- The larger base:
`A₁ = π × R²`

; - The smaller base:
`A₂ = π × r²`

; and - 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.