To find the apothem of a regular octagon (the distance from the center to the midpoint of any side), you can use a specific formula that relates the apothem to the side length of the octagon.

For a regular octagon, the apothem $a$ is given by the formula:

$a = \frac{s}{2 \tan\left(\frac{\pi}{n}\right)}$

Where:

• $s$ is the side length of the octagon.
• $n$ is the number of sides (for an octagon, $n = 8$).
• $\pi$ is approximately 3.1416.

Given that the side length $s = 10$ and $n = 8$, we can substitute these values into the formula:

$a = \frac{10}{2 \tan\left(\frac{\pi}{8}\right)}$

Let’s compute the value of the apothem:

$a = \frac{10}{2 \tan\left(0.3927\right)} \approx \frac{10}{2 \times 0.4142} \approx \frac{10}{0.8284} \approx 12.07$

So, the apothem of the regular octagon is approximately 12.07 units.

Would you like more details on this process or have any other questions? Here are a few follow-up questions:

1. How can the area of a regular octagon be calculated using the apothem?
2. How does the apothem relate to the radius of a circumscribed circle?
3. Can the apothem formula be applied to other polygons like a hexagon or decagon?
4. How would the apothem change if the side length were doubled?
5. What role does the tangent function play in geometry for regular polygons?

Tip: The apothem is especially useful when calculating the area of a regular polygon, as it allows you to use the formula: $\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}$.