What is the relation between the volume of the cylinder and volume of cone having same radius and same height?

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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Picture a rectangle divided into two right triangles by a diagonal. How is the area of the right triangle formed by the diagonal related to the area of the rectangle? The area of any rectangle is the product of its width and length. For example, if a rectangle is 3 inches wide and 5 inches long, its area is 15 square inches (length times width). The figure below shows a rectangle "split" along a diagonal, demonstrating that the rectangle can be thought of as two equal right triangles joined together. The areas of rectangles and right triangles are proportional to one another: a rectangle has twice the area of the right triangle formed by its diagonal.

In a similar way, the volumes of a cone and a cylinder that have identical bases and heights are proportional. If a cone and a cylinder have bases (shown in color) with equal areas, and both have identical heights, then the volume of the cone is one-third the volume of the cylinder.

Imagine turning the cone in the figure upside down, with its point downward. If the cone were hollow with its top open, it could be filled with a liquid just like an ice cream cone. One would have to fill and pour the contents of the cone into the cylinder three times in order to fill up the cylinder.

The figure above also illustrates the terms height and radius for a cone and a cylinder. The base of the cone is a circle of radius r. The height of the cone is the length h of the straight line from the cone's tip to the center of its circular base. Both ends of a cylinder are circles, each of radius r. The height of the cylinder is the length h between the centers of the two ends.

The volume relationship between these cones and cylinders with equal bases and heights can be expressed mathematically. The volume of an object is the amount of space enclosed within it. For example, the volume of a cube is the area of one side times its height. The figure below shows a cube. The area of its base is indicated in color. Multiplying this (colored) area by the height L of the cube gives its volume. And since each dimension (length, width and height) of a cube is identical, its volume is L × L × L, or L 3, where L is the length of each side.

The same procedure can be applied to finding the volume of a cylinder. That is, the area of the base of the cylinder times the height of the cylinder gives its volume. The bases of the cylinder and cone shown previously are circles. The area of a circle is πr 2, where r is the radius of the circle. Therefore, the volume V cyl is given by the equation: V cyl πr 2h (area of its circular base times its height) where r is the radius of the cylinder and h is its height. The volume of the cone (V cone) is one-third that of a cylinder that has the same base and height: .

The cones and cylinders shown previously are right circular cones and right circular cylinders, which means that the central axis of each is perpendicular to the base. There are other types of cylinders and cones, and the proportions and equations that have been developed above also apply to these other types of cylinders and cones.

Philip Edward Koth with

William Arthur Atkins

Bibliography

Abbott, P. Geometry. New York: David Mckay Co., Inc., 1982.

Internet Resources

The Method of Archimedes. American Mathematical Society. <//www.ams.org/new-in-math/cover/archimedes2.html>.

1 Expert Answer

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`.