The given formula for circular segment is:
Ag= r²/2 (Θ/180π-SinΘ)
Ag= r²/2 (Θ/180π-SinΘ)
General Formulas:
Area of a Sector (As) = Θ/360πr²
Area of a Triangle( At)= ½r²sinΘ
As= Ag-At
(area of the segment= sector area - area of the triangle )
Why is the At =1/2 r²sinΘ?
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.
So, r² .
Solution:
Ag = Θ/360πr² - ½ r² sinΘ
I used commutative property to reach the formula for Ag which is
r²/2 (Θ/180π-sinΘ) - 1/2 r² sinΘ + Θ/360πr²
And Using Greatest Common factor
1/2 r²(-sinΘ+ Θ/180π)
Used Commutative property again
1/2 r² (Θ/180π-sinΘ)
Ag = r²/2 (Θ/180π-sinΘ)
And now we derived on the same formula.
QED
Area of a Sector (As) = Θ/360πr²
Area of a Triangle( At)= ½r²sinΘ
As= Ag-At
(area of the segment= sector area - area of the triangle )
Why is the At =1/2 r²sinΘ?
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.
So, r² .
Solution:
Ag = Θ/360πr² - ½ r² sinΘ
I used commutative property to reach the formula for Ag which is
r²/2 (Θ/180π-sinΘ) - 1/2 r² sinΘ + Θ/360πr²
And Using Greatest Common factor
1/2 r²(-sinΘ+ Θ/180π)
Used Commutative property again
1/2 r² (Θ/180π-sinΘ)
Ag = r²/2 (Θ/180π-sinΘ)
And now we derived on the same formula.
QED