For the given diagram, we will find the area of the sector and then the area of the triangle. Next, we will find their difference to find the area of the segment. Finally, we will calculate the difference between the area of the circle and the area of the segment to find the area of the shaded region.
Area of the Sector
The area of a sector of a circle is the product of the measure of the arc divided by 360 and the area of the circle.
With this in mind, let's consider the given diagram.
We can see that the radius of the circle is 12cm. Also, the central angle that corresponds to our arc has a measure of 60. Recall that the measure of an arc is the same as the measure of its corresponding central angle. Therefore, the measure of our arc is also 60. Let's consider how these new pieces of information fit in the diagram.
We have all the information we need to calculate the area of the sector. Let's do it!
The area of a triangle can be found by taking half the product of two side lengths and the sine of the included angle. With this in mind, let's consider the triangle in our diagram.
The length of both of the known sides is 12m and the included angle measures 60^(∘). This is enough information to find the area of the triangle.
Now, to obtain the area of the segment, we need to subtract the area of the triangle from the area of the sector.
Area of the segment = (24 π - 36sqrt(3))m^2
Finally, to find the area of the shaded region, we need to subtract the area of the segment from the area of the circle.
