Why Is My Calculation of a Regular Polygon's Area Incorrect?

[disregardthat]

Hi, I am to find a formula for the area of a regular polygon with a side "a".

I just keep getting the wrong answer: this is how i did it:

if we draw a circle in a coordinate system, with radius "r". The diameter lyes on the x-axis. I draw an angle from the center. This angle is then 360/n where n is the amount of sides the polygon can have.

The two other angles in the triangle we get with two sides "r" and one side "a" is 180/n.

Ok, to find the side r expressed with a:

$$r^2 = r^2 + a^2 2ra \cos{\frac{180}{n}}$$

$$a = 2r \cos{\frac{180}{n}}$$

$$r = \frac{a}{2 \cos{\frac{180}{n}}}$$

The area of this triangle is:

$$A = \frac{1}{2} \sin{\frac{180}{n}} ar = \frac{1}{2} \sin{\frac{180}{n}} \frac{a}{2 \cos{\frac{180}{n}}} a = \frac{1}{4} \tan{\frac{180}{n}} a^2$$

The area of the whole polygon will then be the area of the triangles in the circle. I multiply with the number I divided 360 with, "n".

So: $$A_n =\frac{n}{4} a^2 \tan{\frac{180}{n}}$$

But this is wrong! Why is it wrong?

The correct answer is:
$$A_n =\frac{na^2}{4 \tan{\frac{180}{n}}}$$

 

[Werg22]

$$r^2 = r^2 + a^2 - 2ra \cos{\frac{180}{n}}$$

This is wrong. The angle is 90 - 180/n, hence giving $$r^2 = r^2 + a^2 - 2ra \sin{\frac{180}{n}}$$

 

[uart]

Ok, to find the side r expressed with a:

$$r^2 = r^2 + a^2 2ra \cos{\frac{180}{n}}$$

I've got no idea where that line came from but it looks wrong (edit: ok I now see it was supposed to be the cosine rule). You should have just used :

$$a/2 = r \sin(180/n)$$

Which gives : $$r = \frac{a}{2 \sin(180/n)}$$

Now just substitute that into :

$$A = n ( \frac{1}{2} r^2 \sin(360/n) )$$

PS. Remember to use the trig identity : $$\sin(2x) = 2 \sin(x) \cos(x)$$ if you want to get your answer in exactly the same form as the one given.

 

[disregardthat]

Yes, it was the cosine rule I meant.

Hmm, that was weird. We are not supposed to use trigonometric identities. Or at least the book doesn't mention any of it.

 

[Werg22]

Well, uart expression is equivalent to $$A = n r^2 \sin(180/n)\cos(180/n)$$

 

[uart]

Hmm, that was weird. We are not supposed to use trigonometric identities.

You can get a perfectly good (correct) answer without even using that last trig idenity, it just won't be in the exact same form as the one given. It will be 100% equivalent but just not an identical form.