To find the measure of one interior angle of a regular octagon, we can use the following formula for the measure of an interior angle of a regular polygon with $n$ sides:

$\text{Interior Angle} = \frac{(n - 2) \times 180^\circ}{n}$

For a regular octagon, $n = 8$:

$\text{Interior Angle} = \frac{(8 - 2) \times 180^\circ}{8}$

Simplifying this:

1. Calculate $8 - 2 = 6$.
2. Then, $6 \times 180^\circ = 1080^\circ$.
3. Finally, $\frac{1080^\circ}{8} = 135^\circ$.

So, the measure of one interior angle of a regular octagon is $135^\circ$.

Explanation

This result follows from the fact that a regular polygon’s interior angles are equal, and we can calculate each one by dividing the total sum of the interior angles by the number of sides.

Would you like further details or have any questions?

Here are five related questions to explore further:

1. How would you calculate the sum of the interior angles of an octagon?
2. What is the measure of an exterior angle of a regular octagon?
3. How does the formula change if the polygon is not regular?
4. What is the measure of one interior angle for a regular hexagon?
5. How do the interior and exterior angles relate in any regular polygon?

Tip: For any regular polygon with $n$ sides, the sum of all exterior angles is always $360^\circ$.