New way to derive sectors of a circle (easy)

[shadowboy13]

So for starters the area of an entire circle has 360º,right?

So we can say that: $1∏r^2$ is $\equiv$ to $360º$

So by that logic $0.5∏r^2$ is $\equiv$ to $180º$

And finally $0.25∏r^2$ is $\equiv$ to $90º$

Divide both sides by 9, and you get : $0.25∏r^2/9$ is $\equiv$ to $10º$

From that it's much simpler to multiply both sides by some variable.

Simple right?

 

[pwsnafu]

How is that any different to the formula on Wikipedia?

 

[Mark44]

So for starters the area of an entire circle has 360º,right?

For starters, the area of a circle is not 360°. That's the measure of the angle of a sector.

So we can say that: $1∏r^2$ is $\equiv$ to $360º$

So by that logic $0.5∏r^2$ is $\equiv$ to $180º$

And finally $0.25∏r^2$ is $\equiv$ to $90º$

Divide both sides by 9, and you get : $0.25∏r^2/9$ is $\equiv$ to $10º$

From that it's much simpler to multiply both sides by some variable.

Simple right?

 

[Mentallic]

Try using \pi in your latex code to produce $\pi$ instead of using the product symbol.

If you want to find the area of a sector of a circle that has angle $\theta$ then multiply the area of a circle by $\theta/2\pi$ so

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

However, this assumes that the angle is in radians, but if you want to use degrees instead then just use the conversion

$$\text{angle in radians}=\text{angle in degrees}\times \frac{\pi}{180^o}$$

So the formula is then

$$A=\pi r^2\cdot\frac{\phi}{360}$$

Where $\phi$ is in degrees. So if $\phi=360$ which would be the entire circle, then as expected, you get $A=\pi r^2$

 

[CellCoree]

I appreciate your effort to find a simpler way to derive sectors of a circle. Your approach is certainly creative and could potentially be useful in certain situations. However, I would like to point out that your method assumes that the area of a circle is directly proportional to its angle measure, which is not always the case. In fact, the area of a circle is proportional to the square of its radius, not its angle measure. This means that your equation $1∏r^2$ is not equivalent to $360º$, but rather to $2∏$. Similarly, $0.5∏r^2$ is not equivalent to $180º$, but rather to $∏$. Therefore, your final equation $0.25∏r^2/9$ is not equivalent to $10º$, but rather to $∏/9$. While your approach may work for simple cases where the angle measure and area are directly proportional, it may not hold true for more complex cases. It is important to always consider the underlying principles and assumptions behind any mathematical relationship. Keep exploring and questioning, as that is the essence of science.