1. What is a Regular Octagon?

A regular octagon is an eight-sided polygon where all sides are equal in length and all interior angles are equal. It's a common shape in architecture and design, known for its symmetry and stability.

2. How Does the Calculator Work?

The calculator uses the regular octagon formulas:

Where:

• \( a \) — Apothem (distance from center to midpoint of a side)
• \( \pi \) — Mathematical constant pi (approximately 3.14159)
• \( \tan \) — Tangent trigonometric function

Explanation: The interior angle is constant for all regular octagons, while the side length can be calculated from the apothem measurement.

3. Properties of Regular Octagons

Details: Regular octagons have 8 equal sides, 8 equal interior angles of 135° each, and 8 exterior angles of 45° each. The sum of all interior angles is 1080°.

4. Using the Calculator

Tips: Enter the apothem measurement (distance from center to midpoint of any side) in any consistent units. The calculator will provide the interior angle and side length.

5. Frequently Asked Questions (FAQ)

Q1: What is the apothem of an octagon?
A: The apothem is the distance from the center of the octagon to the midpoint of any side. It's also the radius of the inscribed circle.

Q2: Can I calculate the side length from other measurements?
A: Yes, side length can also be calculated from the circumradius (distance from center to vertex) or from the area of the octagon.

Q3: Are all interior angles really 135°?
A: Yes, in a regular octagon, all interior angles are exactly 135 degrees due to the mathematical properties of regular polygons.

Q4: What's the difference between regular and irregular octagons?
A: Regular octagons have all sides and angles equal, while irregular octagons can have sides and angles of different lengths and measures.

Q5: Where are octagons commonly used?
A: Octagons are used in architecture (stop signs, building designs), engineering, and various decorative applications due to their pleasing symmetry.