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I encountered a question about areas of a circles and sectors on KhanAcademy, I was given the sector area and the central angle of the radian. I know that the ratio of:

$${Area\,\,of\,\, Sector \over Total\,\,area\,\,of\,\,circle} = {Central\,\,angle \over 360deg}$$

But that is the formula to find the sector area. What if I have the sector area and central angle and I have to find the area of the circle?

I Googled and I found that $θ \over 2$ is used to find the area of the circle with given sector area and central angle.

Why is it $θ \over 2$?

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  • $\begingroup$ Using the ratio you mentioned, the sector area can be expressed as $\frac1{2}r^2 \theta$ where $\theta$ is radians. $\endgroup$
    – Nicholas
    Nov 11, 2015 at 10:38

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The formula you saw (perhaps from http://www.purplemath.com/modules/sectors.htm) of $A=\left(\dfrac{\theta}{2}\right) r^2$ might better be written as $A=\frac12\theta r^2$.

The $\frac12$ is similar to that in the area of a triangle being $\frac12 bh$ or in the integral $\displaystyle \int x \, dx = \tfrac12 x^2+c$ and has little to do with half the angle. It results from the sector being wedge shaped rather than almost rectangular (unless you split the sector in two and put it back together with one piece being back to front).

You can check that for a full circle $\theta = 2\pi$ and so the area would be $\frac12 \times 2\pi \times r^2 = \pi r^2$

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  • $\begingroup$ I found this on MathIsFun.com: mathsisfun.com/geometry/iradian-circ.html According to the animation, half a circle is πRadians, so a full circle will be 2 * πRadians. Is that where the formula came from? $\endgroup$ Nov 11, 2015 at 11:42
  • $\begingroup$ @theasianmamba24 The angle of a full circle is $2\pi$ radians (equivalent to $360^\circ$). So your original expression is effectively $\dfrac{A}{\pi r^2} = \dfrac{\theta}{2\pi}$ which reorganising and cancelling $\pi$ gives $A= \frac12 \theta r^2$ $\endgroup$
    – Henry
    Nov 11, 2015 at 12:55
  • $\begingroup$ So, is A the area of the sector? $\endgroup$ Nov 11, 2015 at 13:21
  • $\begingroup$ Yes - as used here $\endgroup$
    – Henry
    Nov 11, 2015 at 13:23
  • $\begingroup$ I understand now! Thanks! $\endgroup$ Nov 12, 2015 at 4:56
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The formula for area of sector is $x/360.πr^2$. And for area of circle you can use the formula you have mentioned above.NOTE x is the central angle.

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