Question

How do I find the area of a shaded region within a circle?

Answer 1

Hint: The given question describes the operation of finding the area of shaded position within a circle using mathematical formulae. Also, remind the general formula to find the area of a circle. Also, it involves the operation of addition/subtraction/multiplication/division. Consider the angle in degree, the distance in\[cm\].

Complete step-by-step solution:
In the given question, we would find how to calculate the area of a shaded region within the circle. For finding the method to find the area of the shaded region in a circle, we assume a diagram as given below,

Here, the origin of the circle is mentioned as “\[o\]” and the radius of the circle is “\[r\]”. Also, the shaded region is marked in the above figure. So, we would find the area of the shaded region. Before that, we would remember the general formula for finding the area of the circle.
The area of circle A \[ = \pi {r^2}\]\[ \to \left( 1 \right)\]
Here \[\pi \] is constant, the value of \[\pi \]is\[3.14\]. And \[r\]is the radius of the circle. The general formula to find the area of the shaded region is given below

The area of the shaded region \[A = \pi \times {r^2} \times \dfrac{{angle}}{{360}}\]
Here the value of \[\pi \]is\[3.14\]and the value of \[r\]is\[5cm\]. The angle is 90 degrees which are given in the diagram. 360 degree is the total angle of a circle.
\[A = \pi \times {r^2} \times \dfrac{{angle}}{{360}}\]
\[A = \pi \times \left( {{5^2}} \right) \times \dfrac{{90}}{{360}}\]
\[
    \\
  A = \left( {3.14} \right) \times 25 \times \dfrac{{90}}{{360}} \\
  A = \left( {3.14} \right) \times 25 \times \dfrac{1}{4} = \dfrac{{78.5}}{4} \\
  A = 19.625c{m^2} \\
 \]
Here the unit of area is\[c{m^2}\].
So, the area of the shaded region within the circle is\[19.625c{m^2}\].

Note: In the question, we would note the angle of the shaded region in degree. If the circle is shaded completely we can use the general formula (A\[ = \pi {r^2}\]) to find the area of the circle. Also, note that the total angle of the circle is\[360\]degrees. Every measurement should be positive.