Question

The circumference of a circle exceeds the diameter by $16.8 cm$. Find the circumference of the circle.

Answer 1

Hint: Circumference of the circle is $2\pi r$ and the diameter is $2r$, here $r$ is the radius of the circle. Use these results to form an equation in $r$ as per the condition given in the question. Find the value of $r$ from this equation and put it back in the circumference formula.

Complete step by step answer:
Let the radius of the circle be $r$ cm.
We know that the circumference of the circle is $2\pi r$ and that its diameter is $2r$.

So according to the condition given in the question we have:

Circumference of the circle exceeds its diameter by 16.8 cm. This can be represented as:
$ \Rightarrow $ Circumference of the circle $ = $ Diameter of the circle $ + $ 16.8
Putting circumference = $2\pi r$ and diameter = $2r$, we’ll get:
$ \Rightarrow 2\pi r = 2r + 16.8$
Taking terms having $r$ on one side of the equation, we’ll get:
$
   \Rightarrow 2\pi r - 2r = 16.8 \\
   \Rightarrow 2r\left( {\pi - 1} \right) = 16.8 \\
$
Now putting $\pi = \dfrac{{22}}{7}$, we’ll get:
$
   \Rightarrow 2r\left( {\dfrac{{22}}{7} - 1} \right) = 16.8 \\
   \Rightarrow 2r \times \dfrac{{15}}{7} = 16.8 \\
$
On cross multiplication we’ll get:
$
   \Rightarrow r = \dfrac{{16.8 \times 7}}{{2 \times 15}} \\
   \Rightarrow r = \dfrac{{117.6}}{{30}} \\
   \Rightarrow r = 3.92 \\
$
Therefore the radius of the circle is 3.92 cm. So, the circumference of the circle is:
$ \Rightarrow {\text{Circumference }} = 2\pi r$
Putting $r = 3.92$ and $\pi = \dfrac{{22}}{7}$, we’ll get:
$
   \Rightarrow {\text{Circumference }} = 2 \times \dfrac{{22}}{7} \times 3.92 \\
   \Rightarrow {\text{Circumference }} = \dfrac{{172.48}}{7} = 24.64 \\
$

Thus, the circumference of the circle is 24.64 cm.

Note: If the relation is given between area and radius or area and circumference of the circle then instead of a linear equation we will get a quadratic equation in r, because area of a circle is $\pi {r^2}$.