The side of a regular decagon is 4 cm
Find the longest possible diagonal of the decagon.

Options

A. `4sqrt(46+20sqrt5) cm`
B. `4sqrt(12sqrt5) cm`
C. `4sqrt(46-20sqrt5) cm`
D. `sqrt(15-2sqrt5) cm`

Answer

To find the longest possible diagonal of a regular decagon with a side length of 4 cm, we can follow these steps:

### Step 1: Understanding the Decagon
A regular decagon has 10 sides. The longest diagonal connects two vertices that are opposite each other. In a decagon, the longest diagonal will connect vertices that are 5 sides apart.

### Step 2: Identify the Diagonal
Let’s label the vertices of the decagon as \( A_1, A_2, A_3, \ldots, A_{10} \). The longest diagonal can be represented as \( A_1A_6 \) (connecting vertex 1 to vertex 6).

### Step 3: Calculate the Length of the Diagonal
To find the length of the diagonal \( A_1A_6 \), we can use the formula for the length of a diagonal in a regular polygon:
\[ d = a \cdot \sqrt{2 - 2 \cos\left(\frac{2\pi k}{n}\right)} \]
where:
- \( d \) is the length of the diagonal,
- \( a \) is the length of a side,
- \( n \) is the number of sides,
- \( k \) is the number of sides between the two vertices connected by the diagonal.

For our decagon:
- \( a = 4 \) cm,
- \( n = 10 \),
- \( k = 5 \) (since \( A_1 \) and \( A_6 \) are 5 sides apart).

### Step 4: Substitute Values into the Formula
Substituting the values into the formula:
\[ d = 4 \cdot \sqrt{2 - 2 \cos\left(\frac{2\pi \cdot 5}{10}\right)} \]
\[ d = 4 \cdot \sqrt{2 - 2 \cos(\pi)} \]
Since \( \cos(\pi) = -1 \):
\[ d = 4 \cdot \sqrt{2 - 2 \cdot (-1)} = 4 \cdot \sqrt{2 + 2} = 4 \cdot \sqrt{4} = 4 \cdot 2 = 8 \text{ cm} \]

### Step 5: Conclusion
The longest diagonal of the decagon is \( 8 \) cm.