Question

A circular plate has a circumference of $25.7$ inches. What is the area of this plate?

Answer 1

Hint: Here we are asked to find the area of a circular plate whose circumference is given. To find the area of the circular plate we first need to find the radius of it by using the formula of the circumference of a circle then we will use it in the formula of area of a circle to find the area of the given circular plate.

Formula Used: The area of a circle whose radius is $r$ : $\pi {r^2}$
The circumference of a circle whose radius is $r$ : $2\pi r$


Complete step by step answer:
It is given that the circumference of a circular plate is $25.7$ inches and we aim to find the area of it.
To find the area of a circle we need to have one component that is the radius of the circle. Here we are not given the radius of a circular plate instead we are given the circumference of the circular plate. So, we will make use of this circumference of the circular plate to find the radius of the plate then we will calculate the area of the circular plate.
We know that the circumference of a circle having a radius $r$ is nothing but $2\pi r$ . And here it is given that the circumference of the circular plate is $25.7$ inches so we can write it as
$2\pi r = 25.7$
Now let us simplify this to find the value of the radius.
$r = \dfrac{{25.7}}{{2\pi }}$
$ \Rightarrow r = \dfrac{{12.85}}{\pi }$
Now we have got the radius of the circular plate as $\dfrac{{12.85}}{\pi }$ inches.
Now let us calculate the radius of the circular plate. We know that the area of a circle having a radius $r$ is given as $\pi {r^2}$ . On substituting the value of the radius in this we get
$\pi {r^2} = \pi {\left( {\dfrac{{12.85}}{\pi }} \right)^2}$
On simplifying this we get
$ \Rightarrow \dfrac{{{{\left( {12.85} \right)}^2}}}{\pi }$
On further simplification we get
$ \Rightarrow {\left( {12.85} \right)^2} \times \dfrac{7}{{22}}$
$ \Rightarrow 165.12 \times \dfrac{7}{{22}}$
$ \therefore A \simeq 52.56$
Hence, the area of the circular plate whose circumference is $25.7$ inches is $52.56$ square inches.

Note: In the above problem we have applied the value of $\pi $ as $\dfrac{{22}}{7}$ . It is our wish to use the fraction value or the decimal value of $\pi $ anyways we will get the same answer. Also, in the final answer, we have rounded off the value to two decimal places. The rounding off of the decimal value is done as follows: if the rightmost digit of the decimal value is greater than or equal to five then it can be dropped off and the preceding digit will be added one if not the digit will just be dropped off without adding one to the preceding digit. This process is carried out till we reach our required decimal place.