To solve for the perimeter of this regular pentagon, we can use the following steps:

Step 1: Understanding the Structure of the Pentagon

Since $O$ is the center of the regular pentagon, the line segment from $O$ to the midpoint of one side (labeled as 12 units) is the apothem of the pentagon. The apothem (a perpendicular distance from the center to the midpoint of a side) can help us calculate the side length of the pentagon.

Step 2: Relationship Between Apothem and Side Length

For a regular pentagon, the apothem ($a$) and side length ($s$) are related by the formula: $a = \frac{s}{2 \tan(\frac{\pi}{5})}$ Given $a = 12$ units, we can rearrange to solve for $s$: $s = 2a \cdot \tan\left(\frac{\pi}{5}\right)$ Substituting $a = 12$: $s = 2 \times 12 \cdot \tan\left(\frac{\pi}{5}\right)$

Step 3: Calculating the Side Length

Using the approximate value of $\tan\left(\frac{\pi}{5}\right) \approx 0.7265$: $s = 2 \times 12 \times 0.7265 \approx 17.44 \text{ units}$

Step 4: Calculating the Perimeter

The perimeter ($P$) of a regular pentagon is given by: $P = 5 \times s$ Substituting $s \approx 17.44$: $P = 5 \times 17.44 \approx 87.2 \text{ units}$

Answer

The perimeter of the pentagon is approximately $87.2$ units.

Would you like further details on any specific part of this calculation?

Here are some related questions you might find helpful:

1. How is the apothem of a regular polygon calculated in terms of side length?
2. What is the relationship between the apothem and the radius of a regular pentagon?
3. Can this method be applied to calculate the perimeter of other regular polygons?
4. How would the perimeter change if the apothem were increased?
5. What are some real-world applications of calculating the perimeter of regular polygons?

Tip: In regular polygons, knowing the apothem can often simplify finding both side lengths and the perimeter.