To solve the problem of finding the apothem of a regular polygon given the area and perimeter, we can use the formula:

$\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}$

Given:

• Area $A = 710.37$ square units
• Perimeter $P = 231.83$ units

We can rearrange the formula to solve for the apothem $a$:

$a = \frac{2 \times \text{Area}}{\text{Perimeter}}$

Substituting the given values:

$a = \frac{2 \times 710.37}{231.83}$

Now let's calculate the apothem.The apothem of the regular polygon is approximately 6.13 units.

Would you like further details or have any additional questions?

Here are five related questions you might find interesting:

1. How does the number of sides of the polygon affect the length of the apothem?
2. What is the relationship between the apothem and the circumradius of a regular polygon?
3. How would the apothem change if the area was doubled while keeping the perimeter constant?
4. Can you derive the formula for the area of a regular polygon using the apothem?
5. How does the apothem help in finding the side length of a regular polygon?

Tip: The apothem is particularly useful in calculating the area of regular polygons because it relates directly to the radius of the inscribed circle.