⭕ Octagon Calculator
Calculate properties of a regular octagon
How to Use This Calculator
Enter Side Length
Input the length of one side of the regular octagon.
Click Calculate
Press the "Calculate Octagon Properties" button to compute all octagon measurements.
Review Results
View the calculated area, perimeter, apothem, and circumradius.
Formula
Area = 2(1 + √2) × s²
Perimeter = 8s
Apothem = s / (2tan(π/8))
Circumradius = s / (2sin(π/8))
Where:
- s = Side length
- √2 ≈ 1.414 (square root of 2)
- Apothem = Distance from center to midpoint of a side
- Circumradius = Distance from center to vertex
Example: Calculate octagon properties for side length = 10
Area = 2(1 + √2) × 10² = 2(1 + 1.414) × 100 = 4.828 × 100 = 482.84 square units
Perimeter = 8 × 10 = 80 units
About Octagon Calculator
The Octagon Calculator helps you find the area, perimeter, apothem, and other properties of a regular octagon. A regular octagon has eight equal sides and eight equal angles (each 135°).
When to Use This Calculator
- Geometry: Calculate octagon properties for geometric problems
- Architecture: Design octagonal buildings, rooms, or structures
- Construction: Plan octagonal patios, decks, or foundations
- Engineering: Design octagonal components and mechanical parts
- Education: Learn and practice octagon mathematics
- Design: Create octagonal patterns, logos, or graphics
Why Use Our Calculator?
- ✅ Complete Properties: Calculates area, perimeter, apothem, and radius
- ✅ Instant Results: Get all octagon measurements immediately
- ✅ Step-by-Step Display: See calculation formulas with your values
- ✅ Works with Any Units: Meters, feet, inches, or any unit
- ✅ 100% Accurate: Precise geometric calculations
- ✅ Completely Free: No registration required
Understanding Regular Octagons
A regular octagon is an eight-sided polygon with all sides equal and all angles equal (135°). Key properties:
- Eight Sides: All sides have equal length
- Eight Angles: Each interior angle = 135° (sum = 1080°)
- Symmetry: Has 8-fold rotational symmetry
- Stop Sign Shape: Regular octagons are used for stop signs worldwide
- Architecture: Common in building design (e.g., octagonal towers, rooms)
- Can be inscribed in a square: Can fit inside a square with vertices on square's edges
Real-World Applications
Stop Signs: Stop signs worldwide use regular octagons. The octagonal shape is highly recognizable and distinguishable from other traffic signs.
Architecture: Octagonal buildings and rooms are common in architecture. Examples include the Tower of the Winds in Athens and many medieval buildings.
Construction: Octagonal gazebos, patios, and decks are popular for their balanced appearance and efficient use of space.
Design: Octagonal tiles, windows, and decorative elements add visual interest while maintaining geometric regularity.
Frequently Asked Questions
What is a regular octagon?
A regular octagon is an eight-sided polygon with all sides equal in length and all interior angles equal to 135°. It has 8-fold rotational symmetry.
How do you calculate octagon area?
For a regular octagon, area = 2(1 + √2) × s², where s is the side length. This simplifies to approximately 4.828 × s².
Why are stop signs octagonal?
Octagonal stop signs are used because the shape is highly recognizable and distinguishable from other traffic signs even when partially obscured or viewed from unusual angles.
What is the interior angle of a regular octagon?
Each interior angle of a regular octagon is 135°. The formula for interior angle is: (n-2) × 180° / n, where n=8, giving (8-2) × 180° / 8 = 135°.
Can octagons tile a plane?
Regular octagons alone cannot perfectly tile a plane. However, octagons can tile with squares, creating a pattern often seen in Islamic art and architecture.
What's the difference between apothem and circumradius?
The apothem is the distance from the center to the midpoint of a side. The circumradius is the distance from the center to a vertex. For octagons, circumradius > apothem.
How many diagonals does an octagon have?
An octagon has 20 diagonals. The formula is n(n-3)/2, where n=8, giving 8(8-3)/2 = 20 diagonals.