
In the formula, the area of a circle is $\pi r^2$, the numerical constant of the expression $\pi r^2$ is?
Answer
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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 ($\pi r^2h$) 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, $A=\dfrac{1}{2} \times 2 \pi r \times r$, 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
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, $A=\dfrac{1}{2} \times 2 \pi r \times r$, 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
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