Derivation of the Volume of a Sphere

To calculate the volume of a sphere, you can use the formula:

$V=\frac{4}{3}\pi r^3$

Where:

• V is the volume of the sphere
• π (pi) is a constant approximately equal to 3.14159
• r is the radius of the sphere

How was the formula derived?

Actually, the formula for the volume of a sphere can be derived using calculus or using geometric method.

Here's an explanation:

1. Imagine a sphere with radius r.
2. Imagine a cylinder with radius r, and height h = r * 2.
3. Place the sphere into the cylinder.

The volume of a cylinder is given by $V_{\text{cylinder}}=\pi r^{2}h$, where r is the radius of the cylinder and h is its height.

The diameter of the sphere is twice the radius, so the height of the cylinder is 2r.

Thus, the volume of the cylinder is $V_{\text{cylinder}}=\pi r^2(2r)=2\pi r^3$.

However, the sphere only fills about two-thirds of this cylinder. So, the volume of the sphere is two-thirds of the cylinder's volume:

$V_{\text{sphere}}=\frac{2}{3}\pi r^3$

And that's where the formula $V=\frac{4}{3}\pi r^3$ comes from, by simply multiplying both sides by $\frac{3}{2}$.

Calculate volume of a sphere