To find the area of a regular octagon, we can use the formula:

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

Step 1: Calculate the perimeter

The perimeter $P$ of a regular octagon is given by the product of the number of sides (8) and the length of one side.

$P = 8 \times \text{side length} = 8 \times 8.3 \, \text{cm} = 66.4 \, \text{cm}$

Step 2: Use the apothem and perimeter in the area formula

We are given that the apothem is 10 cm. Now we can substitute the values into the area formula:

$\text{Area} = \frac{1}{2} \times 66.4 \, \text{cm} \times 10 \, \text{cm} = \frac{1}{2} \times 664 \, \text{cm}^2 = 332 \, \text{cm}^2$

Thus, the area of the regular octagon is 332 cm².

Would you like further details, or do you have any questions?

Here are 5 related questions to explore:

1. How does the apothem relate to the side length of a regular polygon?
2. Can the formula for the area of a regular polygon be applied to other shapes like hexagons or pentagons?
3. How does changing the apothem affect the area of a regular polygon?
4. What is the geometric significance of the apothem in regular polygons?
5. How can the side length be calculated if only the apothem and area are known?

Tip: For regular polygons, the apothem is a helpful tool because it acts like the "height" when calculating the area.