Question

If the diameter of the circle is $5cm$, 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 $5cm$. 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, $2\times \text{Radius}=5cm$
$\Rightarrow \text{Radius}={\displaystyle \frac{5cm}{2}}=2.5cm$
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 $2.5cm$.
By substituting, we get,
$A=\pi r^2$
$\Rightarrow A=\pi {\left(2.5cm\right)}^2$
$\Rightarrow A=6.25\pi c{m}^2$
Therefore the area of a circle with a radius $2.5cm$ 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,
$\Rightarrow A=6.25\times \left({\displaystyle \frac{22}{7}}\right)c{m}^2$
Further simplifying the calculations, we have,
$\Rightarrow A=\left({\displaystyle \frac{137.5}{7}}\right)c{m}^2$
On further simplification and representing it in decimal expression, we have,
$\Rightarrow A=19.643c{m}^2$
Hence the area of a circle whose diameter is $5$ centimetres is $19.643c{m}^2$.

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,
$\Rightarrow C=2\pi \left(2.5cm\right)$
$\Rightarrow C=5\pi cm$

Putting the value of $\pi$, we get,
$\Rightarrow C=\left(5\times {\displaystyle \frac{22}{7}}\right)cm$
$\Rightarrow C={\displaystyle \frac{110}{7}}cm$
$\therefore C=15.715cm$

Therefore, the circumference of a circle with a diameter $5$ centimetre is $15.715cm$.

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 $\left({\displaystyle \frac{22}{7}}\right)$. The radius of a circle is half of the value of the diameter. So, we have, $Diameter=2\times Radius$.