Question

The diameter of a coin is 1cm. If four such coins are placed on a table so that each coin touches the other two coins as shown in the figure then, find the area of the shaded region.
(Take \[\pi =3.14\])

Answer 1

Hint: We solve this problem by using the removal of area technique.
Here the area of the shaded region is given by removing the areas of four quarter circles from the area of the square.
We use the formula of area of square having side length as \['a'\] is given as
\[\Rightarrow A={{a}^{2}}\]
We use the formula of area of quarter circle having the radius as \['r'\]is given as
\[\Rightarrow A=\dfrac{\pi {{r}^{2}}}{4}\]

Complete step by step answer:
We are given that the diameter of circle as
\[\Rightarrow d=1cm\]
We know that the radius of a circle is half of the diameter.
Let us assume that the radius of each circle as \['r'\] then we get
\[\begin{align}
  & \Rightarrow r=\dfrac{d}{2} \\
 & \Rightarrow r=\dfrac{1}{2}cm \\
\end{align}\]
Let us assume that the side length of square ABCD as \['a'\]
Here, we can see that the square is formula by adding the radii of two adjacent circles because we are given that each circle touches the remaining two.
We also know that all the circles are identical then we get the side length of square as
\[\begin{align}
  & \Rightarrow a=2r \\
 & \Rightarrow a=2\times \dfrac{1}{2} \\
 & \Rightarrow a=1cm \\
\end{align}\]
Now, let us assume that the area of square as \[{{A}_{1}}\]
We know that the formula of area of square having side length as \['a'\] is given as
\[\Rightarrow A={{a}^{2}}\]
By using this formula we get the area of square as
\[\begin{align}
  & \Rightarrow {{A}_{1}}={{\left( 1cm \right)}^{2}} \\
 & \Rightarrow {{A}_{1}}=1c{{m}^{2}} \\
\end{align}\]
Now, let us assume that the area of one quarter circle as \[{{A}_{2}}\]
We know that the formula of area of quarter circle having the radius as \['r'\]is given as
\[\Rightarrow A=\dfrac{\pi {{r}^{2}}}{4}\]
By using this formula we get the area of one quarter circle as
\[\begin{align}
  & \Rightarrow {{A}_{2}}=\dfrac{\pi {{\left( \dfrac{1}{2} \right)}^{2}}}{4} \\
 & \Rightarrow {{A}_{2}}=\dfrac{\pi }{16} \\
\end{align}\]
Now, let us assume that the area of shaded region as \[A\]
We know that from the figure that is the area of the shaded region is given by removing the areas of four quarter circles from the area of the square.
By converting the above statement into mathematical equation we get
\[\Rightarrow A={{A}_{1}}-4{{A}_{2}}\]
Now, by substituting the required values in above equation we get
\[\begin{align}
  & \Rightarrow A=1-\left( 4\times \dfrac{\pi }{16} \right) \\
 & \Rightarrow A=1-\dfrac{\pi }{4} \\
\end{align}\]
We are told to take the value of \[\pi \] as 3.14
By substituting \[\pi =3.14\] in above equation we get
\[\begin{align}
  & \Rightarrow A=1-\dfrac{3.14}{4} \\
 & \Rightarrow A=1-0.785 \\
 & \Rightarrow A=0.215 \\
\end{align}\]

Therefore we can conclude that the area of shaded region as \[0.215c{{m}^{2}}\]

Note: Students may make mistakes in calculating the area of a quarter circle.
We are given that the diameter of coin as 1cm then we got the areas of square and quarter circle as
\[\Rightarrow {{A}_{1}}=1c{{m}^{2}}\]
\[\Rightarrow {{A}_{2}}=\dfrac{\pi }{16}\]
But, students may misunderstand the question and assume that the given 1cm is the radius of the coin so that the areas of square and quarter circle will changes to
\[\Rightarrow {{A}_{1}}=4c{{m}^{2}}\]
\[\Rightarrow {{A}_{2}}=\dfrac{\pi }{4}\]
This gives the wrong answer because the given 1cm in the question is diameter not the radius.
So, we need to read the question correctly to solve a problem.