Octagon Calculator
Calculate area, perimeter, diagonals, and other properties of regular octagons
Calculate Octagon Properties
Choose which measurement you know to calculate all other properties
Enter a positive value to calculate all octagon properties
Octagon Properties
Regular Octagon Diagram
Example Calculation
Regular Octagon with Side Length 5 cm
Given: Side length (a) = 5 cm
Area: 2(1 + √2) × 5² = 2(1 + √2) × 25 ≈ 120.71 cm²
Perimeter: 8 × 5 = 40 cm
Long diagonal: 5 × √(4 + 2√2) ≈ 13.07 cm
Medium diagonal: 5 × (1 + √2) ≈ 12.07 cm
Short diagonal: 5 × √(2 + √2) ≈ 9.24 cm
Interior angle: 135°
Octagon Properties
Sides
An octagon has exactly 8 sides
Diagonals
8 short + 8 medium + 4 long diagonals
Interior Angle
Each interior angle = 135°
Sum of Angles
Total interior angles = 1080°
Diagonal Types
Long Diagonal
Connects opposite vertices (4 total)
Medium Diagonal
Connects vertices with 2 between (8 total)
Short Diagonal
Connects vertices with 1 between (8 total)
Real World
Stop signs, tiles, camera apertures
Understanding Regular Octagons
What is a Regular Octagon?
A regular octagon is an 8-sided polygon where all sides have equal length and all interior angles are equal (135°). The name comes from Greek: "Octo" meaning eight and "gon" meaning angle.
Key Properties
- •Sides: 8 equal sides
- •Interior angles: Each 135°, sum = 1080°
- •Diagonals: 20 total (4 long, 8 medium, 8 short)
- •Symmetry: 8-fold rotational and reflectional
Octagon Formulas
Area: A = 2(1 + √2) × a² ≈ 4.828 × a²
Perimeter: P = 8a
Long diagonal: l = a × √(4 + 2√2) ≈ 2.613a
Medium diagonal: m = a × (1 + √2) ≈ 2.414a
Short diagonal: s = a × √(2 + √2) ≈ 1.848a
Note: Where 'a' represents the side length of the regular octagon.
Construction Methods
From Circle
Divide circle into 8 equal parts (45° each), connect adjacent points with straight lines
From Square
Start with square, "chop off" corners at 45° angles to create octagon
Angle Method
Draw sides one by one, each at 135° interior angle from previous side