The side of a regular decagon is 2 cm
Find the apothem of the decagon.

Options

A. `(sqrt(10+2sqrt5))` / (`sqrt5` -1 )cm
B. `2sqrt(5-sqrt5) cm`
C. `2sqrt(5+sqrt5) cm`
D. `sqrt(1+sqrt5) cm`

Answer

To find the apothem of a regular decagon with a side length of 2 cm, we can follow these steps:

### Step 1: Understand the formula for the apothem
The formula for the apothem \( a \) of a regular polygon can be given as:
\[ a = \frac{s}{2 \tan\left(\frac{180}{n}\right)} \]
where \( s \) is the length of a side and \( n \) is the number of sides.

### Step 2: Identify the values
For a decagon:
- The number of sides \( n = 10 \)
- The length of each side \( s = 2 \) cm

### Step 3: Substitute the values into the formula
Substituting \( s \) and \( n \) into the apothem formula:
\[ a = \frac{2}{2 \tan\left(\frac{180}{10}\right)} \]
This simplifies to:
\[ a = \frac{2}{2 \tan(18^\circ)} \]
\[ a = \frac{1}{\tan(18^\circ)} \]

### Step 4: Find the value of \( \tan(18^\circ) \)
Using the known value:
\[ \tan(18^\circ) = \frac{\sqrt{5} - 1}{4} \]
Substituting this value into the equation for \( a \):
\[ a = \frac{1}{\frac{\sqrt{5} - 1}{4}} = \frac{4}{\sqrt{5} - 1} \]

### Step 5: Rationalize the denominator
To rationalize the denominator, multiply the numerator and denominator by \( \sqrt{5} + 1 \):
\[ a = \frac{4(\sqrt{5} + 1)}{(\sqrt{5} - 1)(\sqrt{5} + 1)} = \frac{4(\sqrt{5} + 1)}{5 - 1} = \frac{4(\sqrt{5} + 1)}{4} \]
This simplifies to:
\[ a = \sqrt{5} + 1 \]

### Step 6: Final result
Thus, the apothem of the decagon is:
\[ a = \sqrt{5} + 1 \text{ cm} \]