Question

In the formula, the area of a circle is $\pi r^2$, the numerical constant of the expression $\pi r^2$ is?

Answer 1

Hint: We are given a formula based on the area of a figure and from that we have to identify the numerical constant. A numerical constant is a key number whose value is fixed by an ambiguous definition, often related to by a symbol. (eg. an alphabet letter) or by mathematicians names to facilitate using it across multiple mathematical problems. In this, we have to find a constant having a fixed value. For example in the formula ($\pi r^{2}h$) i.e. the volume of a cylinder. Here radius $r$ is variable and h i.e. height is also variable. It varies from question to question but here $\pi$ is a greek constant having a fixed value of $3.14159$. Here in this formula $\pi$ is the numerical constant. Similarly, we will find the numerical constant in the expression.

Complete step by step answer:
We are given an expression I.e. the area of circle formula $a=\pi r^2$. In this, we have to find the numerical constant. The numerical constant in the formula is the constant having a fixed numerical value.
In the expression $\pi r^2$. Here $r$ is the variable whose values vary according to the geometry of the figure but here $\pi$ is the greek constant whose value is fixed I.e. it is having a fixed value of $3.14159$ which is equal to the ratio of the circumference of any circle to its diameter.

Therefore, the numerical constant in expression $\pi r^2$ is $\pi$.

Additional Information:
One method of deriving this formula, which originated with Archimedes, involving viewing the circle as the limit of the sequence of regular polygons. The area of a regular polygon is half its perimeter multiplied by the distance from its center to its sides, and the corresponding formula – that the area is half the perimeter times the radius - namely, $A={\displaystyle \frac{1}{2}}\times 2\pi r\times r$, holds in the limit for a circle

Note:
In such questions, students mainly get confused by the term numerical constant. The numerical constant is the maths constant having a fixed numerical value. Many mathematical formulas have such constants while many have only variables. The values of variables vary in each question but the value of constant remain constant