Hello, I have this question to which it's solution I'm not quite sure of. The question is based on a diagram, and I can't really present it but I'll explain it to my very best, thank you very much for trying to understand my bad presentation of the question.
The diagram is a circle, with center O and points A and B lying on the circle. Radius is 6cm therefore OA=OB=6cm, such that angle AOB = 79 degrees.
Find the area of the shaded segment of the circle contained between the arc AB and the chord [AB].
The solution was a simple sentence, [ (79degrees/360 degrees) x pi x 6^2 ] - [(1/2)(6)(6)sin79]
I understand that [(1/2)(6)(6)sin79] was derived from the rule 1/2absinc and we're finding the area of the triangle.
Therefore, [ (79degrees/360 degrees) x pi x 6^2 ] must be the area of the sector. However, when I looked up the formula for the area of sector, it's (1/2)(delta)(r)^2, where delta is the angle measured in radians.
Primarily, how did [ (79degrees/360 degrees) x pi x 6^2 ] come about then?
because the [ (79degrees/360 degrees) x pi x 6^2 ] would be the measure of that section of the circle.
the area of the whole circle would be pi x r^2 but since you're only solving for that section it would be 79/360 times that whole area of the circle to solve for that section.
Think of a circular pizza and you were asked to find the area of one slice and there are 10 slices, to find the area of one slice you would multiply the area of the whole pizza by (1/10) which represents one slice.
Hope this helps.
To find the area of the shaded segment, we divide the circle into two parts: the sector, which is the shaded region formed by the arc AB, and the triangle, which is the region formed by the chord AB.
Let's begin by calculating the area of the sector. The formula you mentioned, (1/2)(delta)(r)^2, is indeed the formula for the area of a sector in radians. However, in this case, the angle is given in degrees, so we need to convert it to radians before using the formula.
To convert degrees to radians, you can use the following formula: radians = degrees * (pi/180). In this case, the angle AOB is given as 79 degrees, so we convert it to radians as follows:
radians = 79 * (pi/180).
Now we can substitute this value into the sector formula:
Sector area = (1/2) * (79 * (pi/180)) * (6^2).
Simplifying this expression gives us [(79π/180) * 36] = [(79π * 36)/180] = (79π * 2)/10 = (158π/10) = (15.8π).
So the area of the sector is 15.8π square units.
Next, we calculate the area of the triangle. As you mentioned, we can use the formula for the area of a triangle, which is (1/2) * base * height. The base of the triangle is the chord AB, which has a length of 6 cm. The height can be calculated using the formula height = r * sin(angle), where r is the radius and the angle is half of the angle AOB (since it is an isosceles triangle).
Using angle AOB = 79 degrees, the height can be calculated as follows:
height = 6 * sin(79/2) = 6 * sin(39.5).
Now we can substitute these values into the triangle area formula:
Triangle area = (1/2) * 6 * (6 * sin(39.5)) = 3 * 6 * (6 * sin(39.5)).
Finally, we subtract the area of the triangle from the area of the sector to obtain the area of the shaded segment:
Shaded segment area = Sector area - Triangle area = (15.8π) - (3 * 6 * (6 * sin(39.5))).
Thus, [(79 degrees/360 degrees) * π * 6^2] - [(1/2)(6)(6) * sin(79)] is the correct expression for the area of the shaded segment.