Question

The area of a regular polygon of n sides is (where r is inradius, R is circum radius, and a is side of the triangle)

• A. $\frac{nR^2}{2} sin(\frac{2\pi}{n})$
• B. $nr^2 tan(\frac{\pi}{n})$
• C. $\frac{na^2}{4} cot(\frac{\pi}{n})$
• D. $nR^2 tan(\frac{\pi}{n})$

Answer 1

Answer: (A), (B), (C)

The area of a regular polygon of n sides can be expressed as $\frac{nR^2}{2} sin(\frac{2\pi}{n})$, $nr^2 tan(\frac{\pi}{n})$, or $\frac{na^2}{4} cot(\frac{\pi}{n})$.

Answer 2

The area of a regular polygon can be derived by dividing it into $n$ congruent isosceles triangles. The area of each triangle, and consequently the polygon, can be expressed in terms of the circumradius ($R$), inradius ($r$), or side length ($a$).

1. Using Circumradius ($R$): The area of one isosceles triangle with two sides $R$ and the included angle $\frac{2\pi}{n}$ is $\frac{1}{2}R^2\sin(\frac{2\pi}{n})$. The total area of the polygon is $n$ times this value: Area $= \frac{nR^2}{2}\sin(\frac{2\pi}{n})$.

2. Using Inradius ($r$): In a right-angled triangle formed by the inradius, half the side length ($a/2$), and the circumradius, we have $\tan(\frac{\pi}{n}) = \frac{a/2}{r}$, which gives $a = 2r\tan(\frac{\pi}{n})$. The area of one triangle is $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2}ar$. Substituting $a$, the area of one triangle is $\frac{1}{2}(2r\tan(\frac{\pi}{n}))r = r^2\tan(\frac{\pi}{n})$. The total area of the polygon is $n$ times this value: Area $= nr^2\tan(\frac{\pi}{n})$.

3. Using Side Length ($a$): From the relation $\tan(\frac{\pi}{n}) = \frac{a/2}{r}$, we get $r = \frac{a}{2}\cot(\frac{\pi}{n})$. The area of one triangle is $\frac{1}{2}ar$. Substituting $r$, the area of one triangle is $\frac{1}{2}a(\frac{a}{2}\cot(\frac{\pi}{n})) = \frac{a^2}{4}\cot(\frac{\pi}{n})$. The total area of the polygon is $n$ times this value: Area $= \frac{na^2}{4}\cot(\frac{\pi}{n})$.

The fourth option, $nR^2 \tan(\frac{\pi}{n})$, is not a correct formula for the area of a regular polygon. This can be shown by comparing it to the known formula in terms of $R$ and using trigonometric identities.