Volume of a Sphere

The law for the volume of a sphere is one of the laws that is difficult to prove. We provide you with an online sphere volume calculator, in addition to problems, exercises, and a proof of the sphere volume law using integration. We will talk about the Equation for the volume of a sphere, which is: V = 4/3 πr³

Where the letter V refers to Volume and r is the radius of the sphere, which is the line connecting the center of the sphere to any point on the outer surface of the sphere.

On the side note: Our article on the law of sphere volume is useful for students in the second preparatory or secondary grades or for those who are trying to recall the constants of mathematical laws.

As we know, the unit of volume takes the cubic form because volume is represented in three-dimensional space, and this can be proven by laws. For example, if the radius of the drop is given in cm, the volume will be in cm³. And if it is given in m, the volume will be in m³

Sphere Volume Formula

Before starting with simple problems about the sphere, let’s remember some important and basic laws for the sphere:

• Sphere Volume Formula = Volume = 4/3 πr³
• Sphere Surface Area = Surface Area = 4πr2
• Sphere Circumference = circumference = 2πr

The circumference of a sphere is any circle on the surface of the sphere that has the same center as the sphere. The radius of this circle is equal to the radius of the sphere.

And the diameter of the sphere is the longest straight line connecting two points on the surface of the sphere and passing through its center.

Proof of the Sphere Volume Formula

Let’s try to prove the sphere volume formula through integration, knowing that there are other ways to prove the formula, including through the laws of the cylinder and the cone.

First, let’s assume we have the sphere in the image. And let’s assume the disc (or circle) visible in it is in the form of a flat, horizontal slice of the sphere. Where the radius of the disc is x. And the triangle XZR is a right-angled triangle. As is clear, r represents the radius of the sphere. So, according to the Pythagorean theorem:

x² + z² = r²

And now, by finding x in terms of z and r:

x = √r²-z²

To calculate the area of the circle, we apply the formula for the area of a circle, but the radius here is x, so:

A = π (√r²-z²)² = π (r²-z²)

Now all we need is to find the volume of the sphere, which is clearly the sum of the areas of all the horizontal circles or discs from the bottom to the top. And if we assume the center of the circle is the origin of the coordinates. Then our integration range is from -r to r at the top. That is, we will integrate according to the vertical axis.

How to solve sphere volume problems

After learning the most important laws related to the sphere, anyone can now calculate the volume of a sphere, its area, or any request without resorting to a sphere calculator. We just have to analyze the problem according to the following steps:

1. Analyze all the data or given information of the problem and write it on a separate paper.
2. Study the given information of the problem to know the known and unknown values such as sphere volume, diameter, radius, area, circumference…etc.
3. Find the radius of the circle. If we have the diameter known. We can find the radius by dividing by 2.
4. If we know the surface area of the sphere, we can easily calculate the radius from the sphere surface area formula Surface Area = 4πr2.
5. If we are given the circumference of the sphere, we can calculate the radius from the circumference relation circumference = 2πr.
6. Do not forget to pay attention to the units of the problem, as we must unify them before solving so that no incorrect values are produced.
7. Finally, after collecting all the necessary information, we can apply the sphere formula by substituting the radius and calculating the result.
8. Do not forget to write the unit of volume at the end.

Problems on Sphere Volume

Problem one

Calculate the volume of a sphere with a radius of r = 15cm

Solution: As we mentioned, the formula for the volume of a sphere is given by the following relation:

V = (4/3) πr³
V = (4/3) π(15)³
V = (4/3) x π x 3375
= 14137.1669 cm³

Problem two

Calculate the volume of a sphere if you know that its circumference is 33 inches.

Solution: The circumference of a sphere is a circle. First, we write the circumference formula to deduce what values can be found from it: Sphere Circumference = 2×π×radius = 2πr. We notice that this formula is useful for finding the radius. We substitute the circumference value into the formula (let’s denote the circumference by c):

c = 2πr
33 = 2πr
r = 33/2π
r = 5.2521 Inch

Now that we have calculated the radius, we can easily calculate the volume of the sphere by directly applying the formula and substituting it. (Volume)

Volume = (4/3) πr³
Volume = (4/3) π(5.2521)³
Volume = (4/3) π x 144.8768
Volume = (1.33) x 455.14
Volume = 606.858 Inch³

Problem three

A company manufactured a hollow sphere (hollow from the inside like a soccer ball) and the thickness of the outer shell of the sphere was 0.3 cm. If you know that the total diameter of the sphere is 4 cm. Calculate the volume of the outer shell, the volume of the spherical cavity, and the surface area of the sphere from the inside and outside.

Solution: We must imagine the shape of the hollow sphere before solving. The outer shell of the hollow sphere divides the geometric shape into two overlapping spheres, an inner and an outer. The volume of the outer shell = the volume of the total sphere with the outer surface – the volume of the inner sphere (the spherical cavity).

• The total diameter is 4 cm, so the radius of the total sphere = 4 ÷ 2 = 2 cm.
• We can now calculate the volume of the total sphere Volumeouter = (4/3) πr³ = (4/3) π(2)³ = 33.51 cm3

Now, in order to calculate the volume of the inner sphere, its diameter must be calculated, which = the total diameter of the sphere – the thickness of the outer shell. Therefore:

R’ = 4 – 0.3 = 3.7 cm

Therefore, the radius of the inner sphere is: r' = 3.7/2 = 1.85

By applying the sphere volume formula to the inner sphere:

Volume_inner = (4/3) πr’³
Volume_inner = (4/3) π(1.85)³
Volume_inner = (4/3) π(6.332)
Volume_inner = 26.52 cm³

Which is the volume of the spherical cavity. Therefore, the volume of the outer shell of the sphere is:

Volume_cortex = Volume_outer – Volume_inner
Volume_cortex = 33.51 – 26.52 ≈ 7 cm³

Note that the volume of the shell is approximately one-fifth of the total volume, even though the shell thickness is only 0.3 cm, because the surrounding space occupies a large volume.

To calculate the surface area of the outer sphere, we apply the formula: Surface Area = 4πr2 = 4π(2)2 = 16π cm2

As for the surface area of the inner sphere, we apply the formula to the radius of the inner sphere, which is 1.85, so:

Inner sphere surface area Surface Area = 4πr2 = 4π(1.85)2 ≈ 13.7π cm2