Why is my approach for showing $r^2 \frac{\theta}{2}$ equals the area of a circular sector incorrect? Do we need calculus?

Question

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?

Answer 1

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

Answer 2

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

Answer 3

Your answer is wrong because you multiplied by $\frac{\pi}{180}$ when you should have divided!

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

Answer 4

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.