To find the area of the regular pentagon shown in the image, we can follow these steps:

Step-by-step solution:

1. Recognize the formula for the area of a regular polygon: $\text{Area} = \frac{1}{2} \cdot \text{Perimeter} \cdot \text{Apothem}$

2. Identify the given values:

   • Each side of the pentagon is $10 \, \text{cm}$, so the perimeter is: $\text{Perimeter} = 5 \cdot 10 = 50 \, \text{cm}.$
   • The apothem is provided as $6.9 \, \text{cm}$.

3. Substitute the values into the formula: $\text{Area} = \frac{1}{2} \cdot 50 \cdot 6.9$

4. Calculate the area: $\text{Area} = 25 \cdot 6.9 = 172.5 \, \text{cm}^2.$

Final Answer:

The area of the pentagon is 172.5 cm².

Would you like me to explain any part of this solution in detail or show alternative methods?

Here are some related questions to explore:

1. How is the apothem related to the radius of the pentagon?
2. What is the relationship between the perimeter and the number of sides in any regular polygon?
3. How do you calculate the apothem if it isn’t given explicitly?
4. How would the formula change if this were an irregular polygon?
5. Can the area of the pentagon be derived using triangles instead of the polygon formula?

Tip:

For regular polygons, the apothem acts as the perpendicular height for each triangular section formed by dividing the polygon into equal triangles. Use this property to double-check area calculations!