- #1
- 1,866
- 34
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:
[tex]r^2 = r^2 + a^2 2ra \cos{\frac{180}{n}}[/tex]
[tex]a = 2r \cos{\frac{180}{n}}[/tex]
[tex]r = \frac{a}{2 \cos{\frac{180}{n}}}[/tex]
The area of this triangle is:
[tex]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[/tex]
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: [tex]A_n =\frac{n}{4} a^2 \tan{\frac{180}{n}} [/tex]
But this is wrong! Why is it wrong?
The correct answer is:
[tex]A_n =\frac{na^2}{4 \tan{\frac{180}{n}}}[/tex]
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:
[tex]r^2 = r^2 + a^2 2ra \cos{\frac{180}{n}}[/tex]
[tex]a = 2r \cos{\frac{180}{n}}[/tex]
[tex]r = \frac{a}{2 \cos{\frac{180}{n}}}[/tex]
The area of this triangle is:
[tex]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[/tex]
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: [tex]A_n =\frac{n}{4} a^2 \tan{\frac{180}{n}} [/tex]
But this is wrong! Why is it wrong?
The correct answer is:
[tex]A_n =\frac{na^2}{4 \tan{\frac{180}{n}}}[/tex]
Last edited: