# Truncated Cone Calculator

Table of contents

What is a truncated cone in geometry?Truncated cone formulasHow to use this truncated cone calculator?Related Omni calculatorsFAQsThis truncated cone calculator is a comprehensive tool for solving **various problems related to truncated cones**. Our tool will teach you **how to find the surface area** of a truncated cone: both the lateral area and the total area. Of course, it can compute the **slant height and the volume** as well. Its **best feature** though is that it can **work in reverse**: input whatever data you have and get the missing measurements. No more crying over truncated cones in geometry class. Enjoy!

## What is a truncated cone in geometry?

A truncated cone is a cone with the **apex cut off**. Importantly, the cutting plane is **parallel to the base**, or, equivalently, perpendicular to the height. This solid is also known under the name *frustum*. In everyday life, you can see truncated cones looking at, e.g., lamp shades.

Now, let's move on to discussing truncated cone formulas.

## Truncated cone formulas

First, we need to establish some notation. Look at the picture below. We have:

`R`

for the radius of the base (larger circle);`r`

for the radius of the top surface (smaller circle);`h`

for the height, i.e., the shortest perpendicular distance between the two circles; and`s`

for the slant height, i.e., the distance between the two circles measured along the lateral face.

**Volume formula:**

`V = (1/3) × π × h × (r² + r × R + R²)`

.

**Slant height formula:**

`s = √((R - r)² + h²)`

.

**Lateral surface area formula:**

`Lateral = π × (R + r) × s`

.

**Top and bottom surface area:**

`Top = π × r²`

and `Bottom = π × R²`

.

**Total area formula:**

`Total = Lateral + Top + Bottom`

.

Wow, that's a lot of formulas. But hey, you don't need to learn them by heart! We've implemented them in our truncated cone calculator to save your time and effort.

## How to use this truncated cone calculator?

To use this truncated cone calculator, you only need to enter the data: the remaining fields will get **calculated immediately**. Note that you can **adjust the units** to your needs!

Did you get a **red warning**? This means the data you entered is contradictory - please check it carefully and correct any mistakes.

### What is the slant height of a truncated cone with height 4 and radii 1 and 4?

The answer is **5.** The slant height formula reads:

`s = √((R - r)² + h²)`

.

Plugging in our data, we get:

`s = √((4 - 1)² + 4²) = √25 = 5`

.

### How do I get the slant height formula for a truncated cone?

Use the **Pythagorean theorem** to derive the slant height formula. The **slant height is the hypotenuse**. One of the legs is the difference between the longer and the shorter radius of the truncated cone. The other leg is the height of the truncated cone.

### How do I calculate the lateral surface area of a truncated cone with slant height 4?

You need to know the radii to find the area. If the **radii are 10 and 12**, then the lateral surface area is **276.46**. To derive this answer:

**Add**together the radii:`10 + 12 = 22`

.**Multiply**the result by the slant height:`22 × 4 = 88`

.**Multiply**by`π`

:`88π ≈ 276.46`

.- Remember to include the
**units**: if the dimensions are in cm, then the answer is 276.46 cm².