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Volume of Cone

Volume of Cone

When you look at a volcanic mountain, what does it look like? Perhaps the ice cream cones are more attractive and you may be wondering how much ice cream they can contain.

In this article, you will understand what a cone and conical objects are, how to determine their volume and examples of application of this.

What is a cone?

A cone is a three-dimensional solid that consists of a circular base and a continuous curved surface that tapers to a tip-top called the apex, or vertex.

Some examples of objects with conical shapes are the traffic cone, birthday cone caps, ice cream cones, carrots and so on.

Volume of Cones An image of traffic road cones StudySmarterAn image of traffic road cones - StudySmarter Original

Volume of Cones An image of an ice cream cone StudySmarterAn image of an ice cream cone - StudySmarter Original

Types of cones

There are two basic types of cones:

  • The right circular cones which have their apex perpendicular to or right above the center of their base.

An illustration of a right circular cone - StudySmarter Original

  • The oblique circular cones - do not have their apex above the center of the circular base and as such are not perpendicular to the center.

An illustration of an oblique circular cone - StudySmarter Original

Derivation of the Volume of Cones from the Volume of Cylinders

The volume or capacity of a cone is one-third of its corresponding cylinder. This means that when a cone and a cylinder have the same base dimension and height, the volume of the cone is one-third of the volume of the cylinder. Therefore, a cone can be obtained by dividing a cylinder into three parts. In order to visualize this, we will do a little experiment.

Experiment on deriving cones from cylinders

First, let's understand the following: The volume of a solid can also be understood as its capacity. The capacity of a solid is how much of a liquid – we usually think of water – it can contain.

Now to the experiment. It can be done through the following steps.

Step 1. Get an empty regular cylinder with a known height and radius of the circular base.

Step 2. Get an empty regular cone with its height and radius the same as that of the cylinder.

Step 3. Fill the cone with water to the brim.

Step 4. Pour all the water in the cone into the empty cylinder and observe the level of water in the cylinder.

Note that the cylinder is yet to be filled up.

Step 5. Repeat steps 3 and 4 for the second time on the same cylinder which already has water in it. Notice that the water level rise but the cylinder is yet to be filled to the brim.

Step 6. Repeat steps 3 and 4 for the third time on the same cylinder. Notice that the cylinder is now filled to the brim.

This experiment explains that you would need 3 cones to make a cylinder.

Volume of Cones An illustration on how a cone relates with a cylinder StudySmarterAn illustration on how a cone relates with a cylinder - StudySmarter Original

How to calculate the volume of a cone?

The volume of a cone is the 3D space occupied by it. It is the number of unit cubes that fit in it.

Since we know that a cone is one-third of its corresponding cylinder, it is easier to calculate the volume of a cone.

We recall that the volume of a cylinder is the product of its circular base area and its height and is given by,

VolumeCylinder=πr2×h.

Thus, the volume of a cone is one-third the product of its circular base area and height.

VolumeCone =VolumeCylinder3=πr2×h3

Note that the volume of an oblique cone is calculated with the same formula as that of a regular right cone. This is because of Cavalieri's principle which explains that the volume of a solid shape equals the volume of its corresponding oblique object.

The conical cap has a base radius of 7 cm and a height of 8 cm. Find the volume of the cone. Take π = 3.14.

Solution

We first write out the given values r=7 , h= 8 cm, π=3.14.

The volume of the cone can be calculated by the formula,

VolumeCone=πr2×h3=3.14×72×83=1230.883=410.29 cm3

How to calculate the volume of a cone without a known height?

Sometimes we may be asked to find the volume of cones without a known height. In this case, we need to look out for either a known slant height or a known angle. With a slant height given, you can use Pythagoras' theorem to calculate it. Therefore, let l be the slant height,

l2=r2+h2 h=l2-r2 .

Pythagoras' theorem is only applied when your radius r, slant height l is given but your height is not given.

Volume of Cones All illustration on calculating a cone with an angle subtended at the vertex StudySmarterIllustration of a cone with an angle subtended at the apex - StudySmarter Original

When an angle is given, SOHCAHTOA is applied to derive the value of the height. If the angle at the apex is given, then half of that angle is used to find the height. Thus;

tanθ2=rhh=rtanθ2

where;

θ is the angle at the apex,

l is the slant height of the cone,

h is the height of the cone,

r is the base radius of the cone.

Volume of the Frustum of a cone

The frustum of a cone or a truncated cone has the tip-top (vertex) cut off. It is the shape of most buckets.

Volume of Cones An illustration on how a frustum is formed from a cone StudySmarterAn illustration on how a frustum is formed from a cone - StudySmarter Original

To calculate the volume of a frustum, proportion is used. A frustum has two radii - a radius for the bigger circular surface and another radius for the smaller circular surface.

Volume of Cones An image showing the derivation of volume of a frustrum StudySmarter An image showing the derivation of volume of a frustum - StudySmarter Original

Let R be the radius of the bigger circular surface, and r be the radius of the smaller circular surface. Let hf be the height of the frustum, and H be the height of the cone when complete. And finally, let hc be the height of the small cone which was cut off to form the frustum.

Therefore, while using proportion we have,

Rr=Hhc

The volume of the frustum is equal to the difference between the volume of the big cone and the volume of the small cut cone, and hence we have

Volumefrustum= Volumebig cone - Volumesmall cut cone

But,

Volumebig cone=13πR2H, Volumesmall cone=13πr2hc

Thus, the volume of the frustum is

Volumefrustum=13πR2H-13πr2hc

Calculate the volume of the frustum below

Solution

We write out the given values,

R=20 cm÷2=10 cmr=8 cm÷2=4 cmhf=15 cm

We need to find the height of the full cone H, and the height of the small cone hc;

H=hf+hcH=15+hc

Using proportion;

Rr=Hhc

Remember that;

H=15+hc104 = 15+hchc10×hc = 4 (15+hc) 10hc ==60+4hc10hc-4hc = 606hc = 60 6hc6 =606hc = 10 cm

And so,

H=15+hcH=15+10=25 cmVolume of big cone=13πR2H ==13×227×102×25 =5500021 = 2619.05 cm3

Using proportion;

Rr=Volume of big coneVolume of small cone104=2619.05Volume of small cone10×Volume of small cone=4×2619.05Volume of small cone=4×2619.0510Volume of small cone=1047.62 cm3

Recall that;

Volume of frustum=Volume of big cone -Volume of small coneVolume of frustum=2619.05-1047.62Volume of frustum=1571.43 cm3

A cylinder has a radius 4.2 cm and a height 10 cm. If a conical funnel with the same circular top and height were placed into the cylinder, find the volume of the funnel.

Solution

Volume of a cylinder=πr2hVolume of cylinder=227×4.2×4.2×10Volume of cylinder=554.4 cm3

We are told the cone is of the same height and circular top as that of the cylinder. This means that the volume of the cone is one-third the volume of the cylinder.

Volume of a cone=13×volume of a cylinderVolume of a cone=13×554.4Volume of a cone=184.8 cm3

Volume of Cone - Key takeaways

  • A cone is a three dimensional solid that consists of a circular base and a continuous curved surface which tapers to a tip-top called the apex.

  • The right circular cone and the oblique circular cones are the two types of cones.

  • A cone is one third of a cylinder.

  • The volume of a cone is one-third the circular base area multiplied by its height.

  • The frustum of a cone or a truncated cone has the tip-top (vertex) cut off.

Frequently Asked Questions about Volume of Cone

The volume of a cone is calculated by finding one-third of the product between the circular base area and height.

The formula for the volume of a cone is V = 1/3 × Ab × h, where Ab is the area of the circular base and h is the height.

The volume of a cone can be found without its height when the angle of the apex is known. The slant height makes this angle, and we use simple trigonometry. Also, if the slant height is given, you can derive the height using the Pythagorean theorem.

The volume of a right circular cone is the same as the volume of a regular cone.

Final Volume of Cone Quiz

Question

What is a cone?

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Answer

A cone is a three dimensional solid that consists of a circular base and a continuous curved surface which tapers to a tip-top called the apex.

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Question

How many types of cones are there?

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Answer

2

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Question

What are the types of cone?

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Answer

Right circular and oblique circular cones

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Question

3 cones make a cylinder.

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Answer

True

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Question

What is the frustrum of a cone?

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Answer

This is  cone that has its tip-top (vertex) cut off.

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Question

A bucket takes the shape of a truncated cone. If its 20cm deep with a top radius of 15cm and  base radius of  10cm, calculate the maximum capacity of the bucket to the nearest litres.

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Answer

10 litres

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