Question 164005
--------------------------------------------------------------------------------
Dr.E's Solution:

Hello!

I would stray away from the calculus proofs by integration and other such mathematical inductions until you reach those classes or are told to use such methods in calculus to solve this problem. Since this is an algerbra tutoring website, I doubt the need to do so. Let me know if you need it.

Remember from your earlier studies that the Area of a Circle is = {{{A=pi*r^2}}}

The area of the sector of a circle is proportional its central sector.

In simpler terms, if I said that I wanted you to find {{{3/4}}}ths of the circle, you would simply: {{{A=(3/4)*pi*r^2}}}

Same if I asked you to find {{{1/2}}},  circle: {{{A=(1/2)*pi*r^2}}}

In other terms, we could have said that {{{1/2}}} of the circle, in terms of the central angle, is {{{180^o/360^o}}} or {{{pi/360^o}}} of the (full {{{360^o}}}) {{{A=(180^o/360^o)*pi*r^2}}} or {{{A=(pi/360^o)*pi*r^2}}}

Thus, we can infer that given a central angle labeled {{{BETA}}} , would simply be {{{A=(BETA/360^o)pi*r^2}}}

Put everything in terms of radians ({{{360^o=pi}}}), {{{A=(BETA/2pi)pi*r^2}}}

= {{{A= (1/2)*(BETA/pi)pi*r^2}}} (then the {{{pi}}}'s cancel out)
= {{{A= (1/2)*(BETA)r^2}}}

Thus, by using the sector area formula of a circle, which you can derive from the area of a circle, unless your teacher said you have to derive the area of a circle, which would require your knowledge in integral calculus (which I highly doubt) then this is a valid proof for algebra and geometry.

If you have any more questions, don't hesitate to ask!

Have fun with your algebra!
Dr.E