What is the relationship between the volume of a cone?

Volume of a Cone vs Cylinder

Let's fit a cylinder around a cone.

The volume formulas for cones and cylinders are very similar:

So the cone's volume is exactly one third ( 1 3 ) of a cylinder's volume.

(Try to imagine 3 cones fitting inside a cylinder, if you can!)

Volume of a Sphere vs Cylinder

Now let's fit a cylinder around a sphere .

We must now make the cylinder's height 2r so the sphere fits perfectly inside.

So the sphere's volume is 4 3 vs 2 for the cylinder

Or more simply the sphere's volume is 2 3 of the cylinder's volume!

The Result

And so we get this amazing thing that the volume of a cone and sphere together make a cylinder (assuming they fit each other perfectly, so h=2r):

Isn't mathematics wonderful?

Question: what is the relationship between the volume of a cone and half a sphere (a hemisphere)?

Surface Area

What about their surface areas?

No, it does not work for the cone.

But we do get the same relationship for the sphere and cylinder (2 3 vs 1)

And there is another interesting thing: if we remove the two ends of the cylinder then its surface area is exactly the same as the sphere:

Which means that we could reshape a cylinder (of height 2r and without its ends) to fit perfectly on a sphere (of radius r):

(Research "Archimedes' Hat-Box Theorem" to learn more.)

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A cone is a three-dimensional figure with one circular base. A curved surface connects the base and the vertex.

The volume of a 3 -dimensional solid is the amount of space it occupies.  Volume is measured in cubic units ( in 3 , ft 3 , cm 3 , m 3 , et cetera).  Be sure that all of the measurements are in the same unit before computing the volume.

The volume V of a cone with radius r is one-third the area of the base B times the height h .

V = 1 3 B h     or     V = 1 3 π r 2 h ,     where     B = π r 2

Note : The formula for the volume of an oblique cone is the same as that of a right one.

The volumes of a cone and a cylinder are related in the same way as the volumes of a pyramid and a prism are related. If the heights of a cone and a cylinder are equal, then the volume of the cylinder is three times as much as the volume of a cone.

Example:

Find the volume of the cone shown. Round to the nearest tenth of a cubic centimeter.

Solution

From the figure, the radius of the cone is 8 cm and the height is 18 cm.

The formula for the volume of a cone is,

V = 1 3 π r 2 h

Substitute 8 for r and 18 for h .

V = 1 3 π ( 8 ) 2 ( 18 )

Simplify.

V = 1 3 π ( 64 ) ( 18 ) = 384 π ≈ 1206.4

Therefore, the volume of the cone is about 1206.4 cubic centimeters.

The volume of the cylinder is calculated by using the product of the area of its base by its height. As the base is a cylinder and the formula of its area is `A=pi xx r^2`, all we have to do is to get this number and multiply it by the height of the cylinder. Thus, its volume can be obtained by using the formula `V = pi xx r^2 xx h`. Calculating the volume of the cone is very similar but the final result must be divided by 3. Therefore, the volume of a cone is `V = (pi xx r^2 xx h) : 3`. So, we can take a logical conclusion: “the volume of a cone means the third part of the volume of a cylinder having the same base and the same height”. We can also say that “the volume of a cylinder is the triple of the volume of a cone having the same base and the same height”.

The volume of a body (three-dimensional object) is connected with the capacity that object has to store something like water, air, sand or any other substance. Thus, it seems quite obvious that two solids do not need to be equal (have the same measures) to have the same volume. I can easily build two cardboard boxes having the same shape of a parallelepiped. In spite of having different measures, they have the same volume.

Before giving you the formula, I will start by explaining that a truncated cone is the solid that you obtain when you cut a cone according to a plane parallel to the base, and you forget about the small cone that is formed after that cut. In order to be able to calculate its volume, you will need four measures: the radius of the biggest base (R), the radius of the smallest base (r), the height of the truncated cone (h), the generatrix of the truncated cone (g). After having obtained all these measures you employ the following formula: `V = pi xx h xx (R^2 + R xx r + r^2) : 3`.