Question

What is ratio of the area of a circle to the square of its radius? Also, this is related to the Ratio Topic.

Answer 1

The ratio of the area of a circle to the square of its radius is π.

Step-by-step explanation:

The ratio of the area of a circle to the square of its radius is π. Let's break it down step by step:

1. The area of a circle is given by the formula A = πr², where A is the area and r is the radius.
2. The square of the radius is given by r².
3. So, the ratio of the area of the circle to the square of its radius is A/r² = πr²/r² = π.

Answer 2

The ratio of the area of a circle to the square of its radius is π.

Step-by-step explanation:

The area of a circle is given by the formula A = π * r², where A is the area and r is the radius. The square of the radius is r². To find the ratio of the area to the square of the radius, we can express it as A / r². Substituting the formula for the area, we get π * r² / r². The radius squared in the numerator and denominator cancels out, leaving us with the final ratio of π.

In mathematical terms:

`\[ (A)/(r^2) = (π \cdot r^2)/(r^2) = π. \]`

This result has an elegant geometric interpretation. The ratio π signifies that, regardless of the size of the circle, the area covered inside the circle is always proportional to the square of its radius. This fundamental relationship is a key property of circles and is a constant value, making π a fundamental mathematical constant. The simplicity of the ratio underscores the beauty and universality of mathematical relationships within geometry.

In conclusion, the ratio of the area of a circle to the square of its radius is a concise and precise constant, represented by the symbol π. This fundamental mathematical concept has broad applications in various fields, emphasizing the interconnectedness of geometry and mathematics in describing the physical world.