Area of a Cone Calculator

Welcome to our Area of a Cone Calculator page.

We explain how to find the surface area of a right cone and provide a quick calculator to work it out for you, step-by-step.

The calculator will also find the area of an open cone with no base.

Area of a Cone Calculator

Our calculator finds the surface area of a right circular cone when the radius (or diameter) and height are known.

Area of a Cone Calculator

area of a cone labelled image

Answer1


Answer2

 
 

How the Area of a Cone Calculator Works

  1. Select the type of cone you want: closed or open (default is closed).
  2. Select if you want to use the radius or diameter (default is radius).
  3. For the value, you can choose a whole number, decimal or fraction.
    • You can type a fraction by typing the numerator then '/' then the denominator.
    • You can type a mixed number by typing the whole-number part, then a space then the fraction part.
    • Examples: 2 1/2 (two and one-half); 3 4/5 (three and four-fifths); 7 1/3 (seven and one-third).
  4. Choose your units of measurement (default is none)
  5. Choose your desired accuracy (default is 2 decimal places)
  6. Click the Find Area button
  7. You will be given two answers for the area, one in terms of Pi (π) and the other answer as a decimal value.

Types of Cones

There are many different types of cones, including right cones, oblique cones and elliptic cones.

The cones that are most common, and that we use for our area of a cone calculator, are right circular cones, which is a cone with a circular base whose apex (highest point) is directly above the center of the base.

right circular cone image

Open and Closed Cones

As well as different types of cones, there are also categories of cone within each type: open and closed cones

open and closed cones image

A closed cone has two surfaces which includes a base (or top depending on which way up it is).

An open cone is just a curved surface with no base (or top).

Area of a Cone Formula

open closed cone formulas image

The surface area of a closed cone is: \[ A = \pi r (r + \sqrt (r^2 + h^2)) \]

The surface area of an open cone is: \[ A = \pi r \sqrt (r^2 + h^2) \]

where r is the radius of the circle and h is the perpendicular height of the cone.

Area of a Cone Examples

Area of a Cone Example 1

Find the area of the closed cone below. Give your answer to 1 decimal place.

area of a cone example 1

The height of the cone is 12 cm. The radius is 5 cm.

The formula for the area of a closed cone is: \[ A = \pi r (r + \sqrt (r^2 + h^2)) \]

If we substitute the values of the radius and height into the formala, we get: \[ A = \pi (5) (5 + \sqrt (5^2 + 12^2)) \]

If we simplify this, we get: \[ A = 5 \pi (5 + \sqrt (25 + 144)) = 5 \pi (5 + \sqrt (169)) \]

This means that: \[ A = 5 \pi (5 + 13) = 5 \pi (18) = 90 \pi \]

This gives us a final answer of: \[ A = 282.7 \;\ cm^2 \;\ to \;\ 1 \;\ decimal \;\ place \]

 

Area of a Cone Example 2

Find the surface area of the closed cone below to 1 decimal place.

area of a cone example 2

The cone is lying on its side. The actual height of the cone is 8 inches. The radius is 3 ½ inches.

The formula for the area of a closed cone is: \[ A = \pi r (r + \sqrt (r^2 + h^2)) \]

If we substitute the values of the radius and height into the formala, we get: \[ A = \pi (3 {1 \over 2}) (3 {1 \over 2} + \sqrt ((3 {1 \over 2})^2 + 8^2)) \]

If we simplify this, we get: \[ A = 3 {1 \over 2} \pi (3 {1 \over 2} + \sqrt (12 {1 \over 4} + 64)) = 3 {1 \over 2} \pi (3 {1 \over 2} + \sqrt (76 {1 \over 4})) \]

This means that: \[ A = 3 {1 \over 2} \pi (3 {1 \over 2} + 8.732...) = 3 {1 \over 2} \pi (12.232...) = 42.812... \pi \]

This gives us a final answer of: \[ A = 134.5 \;\ in^2 \;\ to \;\ 1 \;\ decimal \;\ place \]

 

Area of a Cone Example 3

Find the surface area of the closed cone below to 2 decimal places.

area of a cone example 3

The height of the cone is 4.8 meters. The diameter of the cone is 4.6 meters.

To find the radius, we divide the diameter by 2. So the radius = 4.8 ÷ 2 = 2.4 meters.

The formula for the area of a closed cone is: \[ A = \pi r (r + \sqrt (r^2 + h^2)) \]

If we substitute the values of the radius and height into the formala, we get: \[ A = \pi (2.4) (2.4 + \sqrt ((2.4)^2 + (4.6)^2)) \]

If we simplify this, we get: \[ A = 2.4 \pi (2.4 + \sqrt (5.76 + 21.16)) = 2.4 \pi (2.4 + \sqrt (26.92)) \]

This means that: \[ A = 2.4 \pi (2.4 + 5.1884...) = 2.4 \pi (7.588...) = 18.212... \pi \]

This gives us a final answer of: \[ A = 57.22 \;\ m^2 \;\ to \;\ 2 \;\ decimal \;\ places \]

 

Area of a Cone Example 4

Find the surface area of the witch's black hat which is an open cone shown below. Give your answer to 1 decimal place.

area of a cone example 4

The height of the cone is 60 cm. The diameter of the cone is 18 cm.

To find the radius, we divide the diameter by 2. So the radius = 18 ÷ 2 = 9 cm.

The formula for the area of an open cone is: \[ A = \pi r \sqrt (r^2 + h^2) \]

If we substitute the values of the radius and height into the formala, we get: \[ A = \pi (9) \sqrt (9^2 + 60^2) \]

If we simplify this, we get: \[ A = 9 \pi \sqrt (81 + 3600) = 9 \pi \sqrt (3681) \]

This means that: \[ A = 9 \pi (60.671...) = 546.041... \pi \]

This gives us a final answer of: \[ A = 1715.4 \;\ cm^2 \;\ to \;\ 1 \;\ decimal \;\ place \]

 

Surface Area of a Cone Worksheets

We have created two worksheets for you to practice the skill of finding the area of a range of different closed and open cones.

Sheet 1: you have to find the area of different closed cones using the height and radius or diameter measurements.

Sheet 2: you have to find the area of different open and closed cones using the height and radius or diameter measurements.

You can use our area of a cone calculator to check you working out if you get stuck!

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

Volume of a Cone Calculator

We also have a volume of a cone calculator which works in a similar way to the calculator on this page.

More Circle Worksheets and Support Pages

More Area & Volume Calculators

We have a range of other area calculators to help you find the area of a range of different 2d and 3d shapes.

More Area Support and Resources

We have a range of other area worksheets and support pages for a range of different 2d shapes.

Looking for more math calculators?

We have a wide range of free math calculators to help you.

Most of our calculators show you their working out so that you can see exactly what they have done to get the answer.

Our calculator hub page contains links to all of our calculators!

 

How to Print or Save these sheets

how to print information image printer

Need help with printing or saving?
Follow these 3 easy steps to get your worksheets printed out perfectly!

How to Print or Save these sheets

how to print information image printer

Need help with printing or saving?
Follow these 3 easy steps to get your worksheets printed out perfectly!

Math-Salamanders.com

Math Salamanders Main Logo

The Math Salamanders hope you enjoy using these free printable Math worksheets and all our other Math games and resources.

We welcome any comments about our site or worksheets on the Facebook comments box at the bottom of every page.


New! Comments

Have your say about the Math resources on this page! Leave me a comment in the box below.
 

Math Salamanders Shop

math salamanders shop logo

Looking for some fun printable math games?

We have some great games for you to play in our Math Games e-books!

NEW Online Age Calculator

Online Age Calculator image

Do you know how old you are...in weeks? days? hours?

Are you more than a million minutes old?

Find out how old you are to the nearest second!

NEW Fraction Calculators

We have updated and improved our fraction calculators to show you how to solve your fraction problems step-by-step!

Take a look and try them out!

Adding Fractions Calculator image Subtracting Fractions Calculator image Multiply Fractions Calculator image Divide Fractions Calculator image

Some of our TOP pages!

Have a look at some of our most popular pages to see different Math activities and ideas you could use with your child


Support Us

If you are a regular user of our site and appreciate what we do, please consider making a small donation to help us with our costs.

Get a free sample copy of our Math Salamanders Dice Games book with each donation!

FUN Math Games

Math Games Learning Games Captain Salamander

Looking for a fun and motivating way to learn and practice math skills?

Why not try one of our free printable math games with your students!