Calculate volume of a sphere

If you have already calculated the area of a sphere and now you are interested in knowing its volume, below you have the calculator to know the volume of a sphere.

You only have to enter the radius of the sphere and press the calculate button to obtain the volume of this figure. Our tool will provide you with the result in two possible forms of representation so that you can choose the one that is most useful to you:

How is the volume of the sphere of radius r calculated?

For find the volume of a sphereSimply cube the radius. The result you get you have to multiply it by PI and, finally, multiply it by 4/3.

Formula for calculating the volume of a sphere

The theory we have just seen can be summarized in the following mathematical formula that will help you to calculate the volume of a sphere of radius r:

Formula for the volume of a sphere

By example.Let's calculate the volume of a wait with radius = 2 centimeters:

V = 4/3 x π x 23 = 10,66π cubic centimeters = 33, 51 cm3

Note that in the formula we use the radius of the sphere and not its diameter. If the problem statement gives us the diameter, all we have to do is to divide its value by 2:

radius of a sphere = diameter / 2

Once we know how much the radius of the sphere is worth, we apply the formula above.

Demonstration of the volume of a sphere

Area of a sphere calculator

There are several ways to prove the formula for the volume of a sphere. Let's look at the most common ones:

With triple or volume integrals

It is possible to demonstrate the formula for calculating the volume of a sphere using triple integrals or volumetric.

Triple integral

To prove the formula for the volume of a sphere with center at the origin and radius R we know that the points in the first octant can be expressed as:

Sphere coordinates

Therefore, we can solve the triple integral to find the formula for the volume of a sphere as seen here:

Demonstration of the volume of a sphere with triple integrals

Another way to do it with triple integrals:

Triple integral to find the volume of a sphere

With simple integrals

If the triple integrals are too complex for you, there is another way to obtain the demonstration of the volume of a sphere using definite integrals.

To do so, we will start from the equation of the circumference which is this one:

x² + y² = r²

If we rotate a semicircle around the abscissa axis we will have as a result a sphere.

Volume of a sphere

There is an integral that allows us to calculate the volume of the body of revolution that is generated by rotating a curve f(x) around the OX axis and bounded by x = a and x = b although in our case is bounded by -r and r. This integral is as follows:

Volume integral

Which by adapting it to our particular case of the sphere becomes this:

Integral to obtain sphere volume

Calculate the surface of a sphere

If instead of calculating the volume you want to calculate the surface area of a sphere, we also have a calculator for this purpose.

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