Question

If the diameter of the circle is $5\,cm$, what is the circumference and area of the figure?

Answer 1

Hint: Here in this question we want to find the area and circumference of a circle whose diameter is given to us as $5\,cm$. To find the area of a circle, we have a standard formula, $A = \pi {r^2}$. Also, we have a standard formula for finding the circumference of a circle when we are provided with the radius of the circle, $C = 2\pi r$. We know the value of $\pi $ and the value of radius is given to us in the question itself. We substitute known values and determine the area of a circle using the formula.

Complete step by step answer:
The circle is a two dimensional figure and we have to determine the area, where the area is the region or space occupied by the circular field. To determine the area of a circle we have the standard formula $A = \pi {r^2}$ where r represents the radius. In the given question, we are given the length of the diameter in centimetres. So, we will first find the radius using $Diameter = 2 \times Radius$.
So, $2 \times \text{Radius} = 5\,cm$
$ \Rightarrow \text{Radius} = \dfrac{{5\,cm}}{2} = 2.5\,cm$
So, we will get the area of the circle using the formula in the unit square centimetres.

To find the area of a circle, we use formula $A = \pi {r^2}$. The radius of the circle is $2.5\,cm$.
By substituting, we get,
$A = \pi {r^2}$
$ \Rightarrow A = \pi {\left( {2.5\,cm} \right)^2}$
$ \Rightarrow A = 6.25\pi \,c{m^2}$
Therefore the area of a circle with a radius $2.5\,cm$ centimetres is $25\pi \,c{m^2}$.We can substitute the value of $\pi $ to find the area and we can simplify further.Substituting the value of $\pi $, we have,
$ \Rightarrow A = 6.25 \times \left( {\dfrac{{22}}{7}} \right)\,c{m^2}$
Further simplifying the calculations, we have,
$ \Rightarrow A = \left( {\dfrac{{137.5}}{7}} \right)\,c{m^2}$
On further simplification and representing it in decimal expression, we have,
$ \Rightarrow A = 19.643\,c{m^2}$
Hence the area of a circle whose diameter is $5$ centimetres is $19.643\,c{m^2}$.

Now, the circumference of the circle can be calculated using the formula $C = 2\pi r$ by substituting the value of the radius of the circle given in the question itself.
So, we get,
$ \Rightarrow C = 2\pi \left( {2.5\,cm} \right)$
$ \Rightarrow C = 5\pi \,cm$

Putting the value of $\pi $, we get,
$ \Rightarrow C = \left( {5 \times \dfrac{{22}}{7}} \right)\,cm$
$ \Rightarrow C = \dfrac{{110}}{7}\,cm$
$ \therefore C = 15.71\,5\,cm$

Therefore, the circumference of a circle with a diameter $5$ centimetre is $15.71\,5\,cm$.

Note: The circle is a two-dimensional figure and we have to determine the area, where the area is the region or space occupied by the circular field. The area of a circle is defined as the region occupied by the circular region. The radius is denoted by r or R. Value of $\pi $ is approximately taken as $\left( {\dfrac{{22}}{7}} \right)$. The radius of a circle is half of the value of the diameter. So, we have, $Diameter = 2 \times Radius$.