1. What is the Octagon Side Length Formula?

The octagon side length formula calculates the length of each side of a regular octagon when you know the radius (distance from center to vertex). A regular octagon has eight equal sides and eight equal angles.

2. How Does the Calculator Work?

The calculator uses the octagon side length formula:

Where:

• \( s \) — Side length (units)
• \( r \) — Radius (distance from center to vertex, units)
• \( 45^\circ \) — Central angle between two radii (360°/8 sides)

Explanation: The formula derives from the Law of Cosines applied to the isosceles triangle formed by two radii and one side of the octagon.

3. Practical Applications

Details: This calculation is essential in architecture, engineering, and design when creating octagonal structures, frames, or layouts where you need to determine side lengths from a given radius.

4. Using the Calculator

Tips: Enter the radius of your octagon (distance from center to any vertex). The radius must be a positive number. The calculator will output the corresponding side length.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between radius and side length?
A: The radius is the distance from center to vertex, while side length is the distance between two adjacent vertices.

Q2: Can I calculate radius from side length?
A: Yes, the formula can be rearranged: \( r = s / \sqrt{2 - 2 \times \cos(45^\circ)} \)

Q3: How accurate is this calculation?
A: The calculation is mathematically precise for a perfect regular octagon. Real-world measurements may vary slightly.

Q4: Does this work for irregular octagons?
A: No, this formula only applies to regular octagons where all sides and angles are equal.

Q5: What's the simplified version of the formula?
A: Since \( \cos(45^\circ) = \sqrt{2}/2 \), the formula simplifies to \( s = r \times \sqrt{2 - \sqrt{2}} \)