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I know that the area of the sector of circle can be computed using the following two formulas

$$\pi r^2 \frac{\theta }{360} \space \space \text{ (degrees case)}$$ $$or$$ $$r^2 \frac{\theta }{2} \space \space \text{ (radians case)}$$

The first case makes complete sense. You just consider the fraction of the full 360 degree circle and use this fraction to scale the full area formula.

The second case doesn't make much sense. I've searched for proofs and most relied on calculus, which seems like overkill... Is there a simpler way of showing that the radian case is valid?

As an attempt I tried converting the degrees to radians using $\frac{\pi}{180}$

$\pi r^2\frac{\theta \space \cdot \frac{\pi}{180}}{360}= \pi^2 r^2 \frac{\theta}{64800} \neq r^2 \frac{\theta}{2}$

Question: Why is my approach for converting the degree case of the formula to the radian case of the formula incorrect?

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  • $\begingroup$ "The second case doesn't make much sense.": why not ?? $\endgroup$
    – user65203
    May 13 '19 at 9:07
  • $\begingroup$ @YvesDaoust Let's just say it doesn't make as much intuitive sense, but if you have any intuition for why a radian measure divided by 2 multiplied by the radius squared is the area of a sector please let me know $\endgroup$
    – ApprenticeOfMathematics
    May 13 '19 at 9:11
  • $\begingroup$ What's your intuition in the case of degrees ? $\endgroup$
    – user65203
    May 13 '19 at 9:12
  • $\begingroup$ From the book 'The Joy of X': Suppose the length of a hallway is $y$ when measured in yards and $f$ is measured in feet. Then the formula is not $y = 3f$. The correct formula is actually $f = 3y$ (try it!), because $3$ actually means $3$ feet per yard. When you multiply it by $y$, the units cancel out, and you are left with units of feet. $\endgroup$
    – Toby Mak
    May 13 '19 at 9:20
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The area of a full circle is $$\pi r^2$$

and you are taking a fraction

$$\frac\theta{360}$$ when the angle is expressed in degrees or

$$\frac\theta{2\pi}$$ when it is in radians.

Or $$\frac\theta{400}$$ when it is in gradians (grades) or $$\theta$$ in revolutions (turns) or $$\frac\theta4$$ in quadrants or $$\frac\theta{256}$$ in binary radians.

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  • $\begingroup$ +1 for intuition, i see now that the 2$pi$ in the radian case is cancelled $\endgroup$
    – ApprenticeOfMathematics
    May 13 '19 at 9:21
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If we convert degrees to radians, we multiply by $\frac{\pi}{180}$.

So, $\theta^o \frac{\pi}{180} = \theta^c$ [c - radians, o - degress]

We have $\theta^o$, so $\theta^o = \theta^c \frac{180}{\pi}$

So, $$A = \pi r^2\frac{\theta^o}{360} = \pi r^2 \frac{180}{\pi} \frac{\theta^c}{360} $$

or $$A = r^2\frac{\theta^c}{2}$$

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  • $\begingroup$ Why $c$ for radians ? $\endgroup$
    – user65203
    May 13 '19 at 9:22
  • $\begingroup$ I'm assuming he just used the superscripts to differentiate a degree measure from a radian measure $\endgroup$
    – ApprenticeOfMathematics
    May 13 '19 at 9:24
  • $\begingroup$ And we generally use c or rad for radians in Math. en.wikipedia.org/wiki/Radian $\endgroup$
    – Ak.
    May 13 '19 at 9:25
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Your answer is wrong because you multiplied by $\frac{\pi}{180}$ when you should have divided!

$\pi r^2\frac{{\theta \div \frac{\pi}{180}}}{360}= \pi r^2 \frac{180 \theta} {\pi 360} = r^2 \frac{\theta}{2}$

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To convert $\frac{\theta}{360º}$ into radians, $\theta$ is a variable and can be either in degrees or radians: $\theta = 2\pi$ or $\theta = 360º$ both work. However, if $\theta$ is in radians, $360º$ should also be in radians, or otherwise you have to multiply by some conversion factor.

Therefore we have: $$A = \pi r^2 \frac{\theta}{2\pi}$$ $$= r^2 \frac{\theta}{2}$$

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  • $\begingroup$ My post has been edited. Can the downvoter please retract their downvote? $\endgroup$
    – Toby Mak
    May 13 '19 at 9:17

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