The question asks about the significance of $\pi$ in mathematics. The options are: a) It is the ratio of a circle's circumference to its diameter. b) It is the length of a circle's radius. c) It is the volume of a sphere.

1. Problem Description

The question asks about the significance of $\pi$ in mathematics. The options are:

a) It is the ratio of a circle's circumference to its diameter.

b) It is the length of a circle's radius.

c) It is the volume of a sphere.

2. Solution Steps

We need to determine which of the given options accurately describes the significance of $\pi$.

The circumference of a circle is given by $C = 2\pi r$, where $r$ is the radius of the circle. Since the diameter $d = 2r$, we can also write the circumference as $C = \pi d$.

From this formula, we can express $\pi$ as the ratio of the circumference to the diameter:

$\pi = \frac{C}{d}$.

The value $\pi$ is not simply the length of a circle's radius.

The volume of a sphere is given by $V = \frac{4}{3}\pi r^3$. So $\pi$ is used to calculate the volume of a sphere, but the significance of $\pi$ is not solely its use in the volume of a sphere.

Therefore, $\pi$ is the ratio of a circle's circumference to its diameter.

3. Final Answer

It is the ratio of a circle's circumference to its diameter.