To determine the apothem of the regular octagon, we can use the formula for the area of a regular polygon:

$A = \frac{1}{2} \times P \times a$

Where:

• $A$ is the area of the polygon.
• $P$ is the perimeter of the polygon.
• $a$ is the apothem (what we need to find).

Given:

• The area $A = 10.15 \, \text{cm}^2$.
• Each side length $s = 1.45 \, \text{cm}$ (as shown in the image).
• The polygon has 8 sides (since it's an octagon).

Step 1: Find the perimeter $P$.

The perimeter $P$ of a regular polygon is given by:
$P = n \times s$
Where $n = 8$ (number of sides) and $s = 1.45 \, \text{cm}$.

$P = 8 \times 1.45 = 11.6 \, \text{cm}.$

Step 2: Solve for the apothem $a$.

Rearrange the area formula to isolate $a$:

$a = \frac{2A}{P}.$

Substitute the known values:

• $A = 10.15$
• $P = 11.6$

$a = \frac{2 \times 10.15}{11.6}.$

$a = \frac{20.3}{11.6} \approx 1.75 \, \text{cm}.$

Final Answer:

The measure of the apothem, rounded to the nearest hundredth, is 1.75 cm.

Do you need a further explanation or breakdown? Let me know!

Here are 5 related questions:

1. How do you calculate the perimeter of other regular polygons?
2. What is the formula for the area of a regular polygon with any number of sides?
3. How does the apothem relate to the radius of a regular polygon?
4. Can you find the area of a regular polygon if only the apothem and perimeter are given?
5. How does increasing the number of sides affect the apothem of a regular polygon?

Tip: Always remember that the apothem is the perpendicular distance from the center to the midpoint of any side of a regular polygon. It acts like a height in the area calculation.