Question

The inner and outer diameters of ring I of a dartboard are 32cm and 34cm respectively and those of ring II are 19cm and 21cm respectively. What is the total area of these two rings?

Answer 1

Hint: For solving this type of question you should know about the area of rings and in this question, it is calculated by subtracting the area of small from the area of the big radius circle. Though it will be the result of an area of one ring. Then do the same with the second ring and when you add both the areas of rings then the final area or total area of rings can be determined.

Complete step by step answer:
In the question it is given that there are two rings which are ring I and ring II and the diameters of both rings of a dartboard are 32cm and 34cm and 19cm and 21cm respectively. So, we can write it as follows,
For ring I $\Rightarrow {{r}_{1}}=16,{{r}_{2}}=17$
And for ring II$\Rightarrow {{r}_{1}}=9.5,{{r}_{2}}=10.5$
Since we know that radius = $\dfrac{\text{diameter}}{2}$.
According to the question,
Area of ring I $=\pi {{r}_{2}}^{2}-\pi {{r}_{1}}^{2}\Rightarrow \pi \left( {{r}_{2}}^{2}-{{r}_{1}}^{2} \right)$
By substituting the values in the formula, we get as,
$\begin{align}
  & \text{Are}{{\text{a}}_{1}}=\dfrac{22}{7}\left( {{17}^{2}}-{{16}^{2}} \right)c{{m}^{2}} \\
 & \Rightarrow \text{Are}{{\text{a}}_{1}}=\dfrac{22}{7}\left( 17+16 \right)\left( 17-16 \right)c{{m}^{2}} \\
 & \Rightarrow \text{Are}{{\text{a}}_{1}}=\dfrac{22}{7}\times 33c{{m}^{2}} \\
\end{align}$
And the area of ring II is,
$\text{Are}{{\text{a}}_{2}}=\pi \left( {{r}_{2}}^{2}-{{r}_{1}}^{2} \right)$
By substituting the values in the formula, we get as,
$\begin{align}
  & \text{Are}{{\text{a}}_{2}}=\dfrac{22}{7}\left( {{\left( 10.5 \right)}^{2}}-{{\left( 9.5 \right)}^{2}} \right)c{{m}^{2}} \\
 & \Rightarrow \text{Are}{{\text{a}}_{2}}=\dfrac{22}{7}\left( 10.5+9.5 \right)\left( 10.5-9.5 \right)c{{m}^{2}} \\
 & \Rightarrow \text{Are}{{\text{a}}_{2}}=\dfrac{22}{7}\times 20c{{m}^{2}} \\
\end{align}$
So, the total area of the two rings = Area of ring I + Area of ring II. So, total area is,
$\begin{align}
  & \dfrac{22}{7}\times 33+\dfrac{22}{7}\times 20 \\
 & \Rightarrow \dfrac{22}{7}\left( 33+20 \right)=166.57c{{m}^{2}} \\
\end{align}$

So, the total area of both the rings are $166.57c{{m}^{2}}$.

Note: You should be careful when taking the values of radius because it will affect your area. So, always remember that the bigger diameter is of the outer side and the smaller one is of the inside diameter. So, always subtract the small area from the bigger area. And all the values should be in the same form as cm or m etc.