Volume of a Sphere

The volume of a sphere is the space it occupies in the three-dimensional plane. It refers to the volume of a solid sphere. The volume is also the measure of the capacity of a sphere or the number of unit cubes that can be fit into it. It is measured in cubic units such as m³, cm³, mm³, ft³.

Let us learn how to find the volume of a solid sphere.

Formulas

With Radius

The basic formula to calculate the volume of a sphere is:

Let us now learn how to derive the above formula.

Derivation

1. Using Integration

Let us consider a sphere formed with a number of thin circular discs stacked one above the other, as shown in the figure below.

The diameters of the discs are continuously varying. The center of all the discs is collinear.

Now, we choose any one disc with a radius ‘x’ and thickness ‘dy’. The disc is at a distance ‘y’ from the x- axis.

Thus, the volume can be written as:

Volume = Area of the circular disc × Thickness of the circular disc

Also, radius of the disc ‘x’ can be expressed in terms of the vertical dimension ‘y’ applying the Pythagorean Theorem.

∴The volume of the disc element ‘dV’ can be written as:

dV = (πx²)dy

Now, x² = r² – y² (Pythagorean theorem)

∴dV = π(r² – y²)dy

Thus, by integrating the above equation, the total volume can be written as:

${V=\int ^{y= +r}_{y= -r}dV}$

${V=\int ^{y= +r}_{y= -r}\pi \left( r^{2}-y^{2}\right) dy}$

${V=\pi \left[ r^{2}y-\dfrac{y^{3}}{3}\right] _{y=-r}^{y=+r}}$

Now, substituting the limits:

${V=\pi \left[ \left( r^{3}-\dfrac{r^{3}}{3}\right) -\left( -r^{3}+\dfrac{r^{3}}{3}\right) \right]}$

Simplifying the above equation:

${V=\pi \left[ 2r^{3}-\dfrac{2r^{3}}{3}\right]}$

${V=\dfrac{\pi }{3}\left[ 6r^{3}-2r^{3}\right]}$

${V=\dfrac{\pi }{3}\left( 4r^{3}\right)}$

 ∴ The final dimensional formula of volume of a sphere is:

Volume (V ) = ${\dfrac{4}{3}\pi r^{3}}$

2. Using Volume of a Cylinder and Cone

As explained by Archimedes, a cylinder, cone, and sphere with a radius ‘r’, and equal cross-sectional area, have their volumes in the ratio of 1:2:3.

Therefore, the relation between the volume of sphere, cone, and cylinder is:

Volume of a Cylinder = Volume of a Cone + Volume of a Sphere

As we know,

Volume of a Cylinder = Volume of a Cone + Volume of a Sphere

∴Volume (V) of a Sphere = Volume (V) of a Cylinder – Volume (V) of a Cone
As we know,

V_cylinder = πr²h

And,

V_cone = 1/3 of V_cylinder = (1/3)πr²h
Now,

V_Sphere = V_cylinder – V_cone
∴ V_Sphere = πr²h – (1/3)πr²h = (2/3)πr²h
here, height of the cylinder = diameter of the sphere = 2r

∴ V_Sphere = (2/3)πr² × 2r

Volume (V ) = (4/3)πr³

Let us solve an example to involving the above formula.

Let us find the volume of a sphere when the radius is not given directly.

With Diameter

The formula to find the volume of a sphere using diameter is:

Let us solve an example to involving the above formula.

We have learned how to find the volume of a solid sphere. Now let us learn the formula to find the volume of a hollow sphere.

Volume of a Hollow Sphere

When a small, inner sphere is removed from a solid sphere, a spherical cavity is formed. The cavity creates a uniform thickness around itself with respect to the solid portion of the sphere. Such a sphere is a hollow sphere. The cavity has the same center as that of the solid sphere. A coconut is somewhat similar to a hollow sphere though it is not a perfect sphere.

Let us now derive the formula for finding the volume of a hollow sphere.

Let,

Radius of outer sphere = R

Radius of inner sphere = r

∴ R > r

So, the volume of such a sphere is expressed as:

Volume of the hollow sphere is = Volume of the outer sphere – Volume of the inner sphere

Volume of the outer sphere = 4/3πR³ 

Volume of the inner sphere = 4/3πr³ 

So, the formula is given below:

Let us solve an example involving the above formula.