Question

Four equal circles each of radius \[a\] units touch one another. The area enclosed between them in square units is
(a) \[3{a^2}\]
(b) \[\dfrac{{6{a^2}}}{7}\]
(c) \[\dfrac{{41{a^2}}}{7}\]
(d) \[\dfrac{{{a^2}}}{7}\]

Answer 1

Hint: We need to find the area enclosed between the four circles. The line segments joining the centres of adjacent circles forms a square. We will find the area of square and area of quadrant of a circle using the respective formulas. Using their values, we will calculate the area of the region enclosed between the four circles.

Formula Used:
We will use the following formulas:
1.The area of a square is given by the formula \[{s^2}\], where \[s\] is the length of the side of the square.
2.The area of the quadrant of a circle is given by the formula \[\dfrac{1}{4}\pi {r^2}\], where \[r\] is the radius of the circle.

Complete step-by-step answer:
First, we will draw the figure from the given information.

Here, A, B, C, and D are the four centres of the four equal circles
We need to find the area of the shaded region.
The radii of the circles with centres A and D form the line segment AD.
Similarly, we get the line segments BC, AB, and CD.
The line segment AD is the sum of the radius of the two circles with centres A and D.
Therefore, we get
\[AD = a + a = 2a\]
Similarly, we get
\[AB = BC = CD = 2a\]
We can observe that the quadrilateral formed by the four line segments is the square ABCD.
Now, we will find the area of the square ABCD.
The area of a square is given by the formula \[{s^2}\], where \[s\] is the length of the side of the square.
Substituting \[s = 2a\] in the formula for area of a square, we get
Area of square ABCD \[ = {\left( {2a} \right)^2}\]
Applying the exponent on the base, we get
\[ \Rightarrow \] Area of square ABCD \[ = 4{a^2}\]
Now, we will find the area of the quadrant of the circle with centre A.
The area of the quadrant of a circle is given by the formula \[\dfrac{1}{4}\pi {r^2}\], where \[r\] is the radius of the circle.
Substituting \[r = a\] in the formula for area of quadrant, we get
Area of quadrant of circle with centre A \[ = \dfrac{1}{4}\pi {a^2}\]
Since the four circles are equal, the area of their quadrants is also equal.
\[ \Rightarrow \] The area of the quadrant of all the four circles is \[\dfrac{1}{4}\pi {a^2}\].
From the figure, we can observe that the area enclosed by the four circles is equal to the difference in the area of the square ABCD, and the quadrants of the four circles.
Therefore, we get
Area enclosed by the four equal circles \[ = 4{a^2} - \dfrac{1}{4}\pi {a^2} - \dfrac{1}{4}\pi {a^2} - \dfrac{1}{4}\pi {a^2} - \dfrac{1}{4}\pi {a^2}\]
Converting subtraction to addition using parentheses, we get
\[ \Rightarrow \] Area enclosed by the four equal circles \[ = 4{a^2} - \left( {\dfrac{1}{4}\pi {a^2} + \dfrac{1}{4}\pi {a^2} + \dfrac{1}{4}\pi {a^2} + \dfrac{1}{4}\pi {a^2}} \right)\]
Adding the terms in the parentheses, we get
\[ \Rightarrow \] Area enclosed by the four equal circles \[ = 4{a^2} - \left( {\dfrac{4}{4}\pi {a^2}} \right) = 4{a^2} - \pi {a^2}\]
Substituting \[\pi = \dfrac{{22}}{7}\] in the expression, we get
\[ \Rightarrow \] Area enclosed by the four equal circles \[ = 4{a^2} - \dfrac{{22}}{7}{a^2}\]
Subtracting the terms in the expression, we get
\[ \Rightarrow \] Area enclosed by the four equal circles \[ = \dfrac{{28 - 22}}{7}{a^2} = \dfrac{{6{a^2}}}{7}\]
Therefore, we get the area of the enclosed area as \[\dfrac{{6{a^2}}}{7}\].
Thus, the correct option is option (b).
Note: Here, we can make a mistake by writing the answer as the area of the square. This is incorrect, because the quadrants are a part of the circles. Therefore, they are not to be included in the area of the region enclosed by the four circles. Square is a two dimensional figure which has 4 equal sides. Circle is a two dimensional figure with no edges and no vertices.