The area of a regular polygon of `n` sides is (where `r` is inradius, `R` is circumradius, and `a` is side of the triangle) `(n R^2)/2sin((2pi)/n)` (b) `n r^2tan(pi/n)` `(n a^2)/4cotpi/n` (d) `n R^2tan(pi/n)`

Options

A. `(nR^(2))/(2) sin ((2pi)/(n))`
B. `nr^(2) tan ((pi)/(n))`
C. `(na^(2))/(4) cot.(pi)/(n)`
D. `nR^(2) tan((pi)/(n))`

Explanation

To solve the problem, we need to determine the area of a regular polygon with \( n \) sides in terms of its inradius \( r \), circumradius \( R \), and side length \( a \). The options provided are:

1. \( \frac{n R^2}{2 \sin\left(\frac{2\pi}{n}\right)} \)
2. \( n r^2 \tan\left(\frac{\pi}{n}\right) \)
3. \( \frac{n a^2}{4 \cot\left(\frac{\pi}{n}\right)} \)
4. \( n R^2 \tan\left(\frac{\pi}{n}\right) \)

### Step-by-Step Solution:

**Step 1: Understand the relationship between the inradius, circumradius, and side length.**

For a regular polygon:
- The side length \( a \) can be expressed in terms of the circumradius \( R \) as:
\[ a = 2R \sin\left(\frac{\pi}{n}\right) \]

**Hint:** Remember that the circumradius \( R \) relates to the side length \( a \) through the sine function.

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**Step 2: Calculate the area of the polygon using the side length.**

The area \( A \) of a regular polygon can be calculated using the formula:
\[ A = \frac{n a^2}{4 \tan\left(\frac{\pi}{n}\right)} \]

**Hint:** The area formula for a polygon is based on the number of sides and the square of the side length.

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**Step 3: Substitute the expression for \( a \) into the area formula.**

Substituting \( a = 2R \sin\left(\frac{\pi}{n}\right) \) into the area formula:
\[ A = \frac{n (2R \sin\left(\frac{\pi}{n}\right))^2}{4 \tan\left(\frac{\pi}{n}\right)} = \frac{n \cdot 4R^2 \sin^2\left(\frac{\pi}{n}\right)}{4 \tan\left(\frac{\pi}{n}\right)} = n R^2 \frac{\sin^2\left(\frac{\pi}{n}\right)}{\tan\left(\frac{\pi}{n}\right)} \]

**Hint:** When substituting, make sure to simplify the expression correctly.

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**Step 4: Use the identity \( \tan(x) = \frac{\sin(x)}{\cos(x)} \) to simplify further.**

Using the identity for tangent:
\[ A = n R^2 \frac{\sin^2\left(\frac{\pi}{n}\right)}{\frac{\sin\left(\frac{\pi}{n}\right)}{\cos\left(\frac{\pi}{n}\right)}} = n R^2 \sin\left(\frac{\pi}{n}\right) \cos\left(\frac{\pi}{n}\right) \]

**Hint:** Remember to apply trigonometric identities to simplify expressions.

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**Step 5: Relate the area to the inradius \( r \).**

The area can also be expressed in terms of the inradius \( r \):
\[ A = \frac{n r^2}{\tan\left(\frac{\pi}{n}\right)} \]

**Hint:** This relationship shows how the area can be expressed using the inradius.

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**Step 6: Conclusion and answer selection.**

From our calculations, we have derived two valid expressions for the area of a regular polygon:
1. \( A = n R^2 \tan\left(\frac{\pi}{n}\right) \)
2. \( A = n r^2 \tan\left(\frac{\pi}{n}\right) \)

Thus, the correct options are:
- \( n r^2 \tan\left(\frac{\pi}{n}\right) \) (Option b)
- \( n R^2 \tan\left(\frac{\pi}{n}\right) \) (Option d)

**Final Answer:** Options (b) and (d) are correct.

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