Derivation of the Line Segment.
Here are the steps to find the formula of the area of the segment or (or we can say Ag) if it is in the simplest form.
The 1st thing to do to find the formula of the area of the segment which is (Ag) is that we can use the formula which is Area of the sector (As) minus the Area of the Triangle (At) (As-At=Ag)
* Our formula for line segment or (Ag) is R^2/2(Θ/180π-sinΘ)
How do we get to this formula?
- We Let As=Θ/360πr^2 and Let At=1/2 r^2 sinΘ
And why is the Area of the triangle (At) 1/2 r^2 sinΘ ?
- because the formula for finding the area of the triangle given 2 sides and an included angle is 1/2ab * sin C but since the given is an isosceles triangle (both sides are equal) then a=b=r
hence, r^2.
then we start solving it
Ag= Θ/360πr^2 - 1/2r^2 sin Θ
And then we are gonna use commutative property to reach the formula for Ag which is:
R^2/2(Θ/180π-sinΘ) - 1/2 r^2 sin Θ + Θ/360πr^2
Using Greatest Common factor..
1/2 r^2 ( -sinΘ+ Θ/180π)
then we Used Commutative property again
1/2 r^2 (Θ/180π-sinΘ)
= r^2/2 (Θ/180π-sinΘ)
finally we derived from the same formula (which is the line segment (Ag))