Octagon Calculator
Calculate area, perimeter, inradius, circumradius, and diagonals of a regular octagon
Length of one side of the regular octagon
Total area of the octagon in square units
Sum of all eight sides
Octagon Properties
Side Length
Perimeter
Area
Inradius (Apothem)
r = (s/2)(1 + √2)
Circumradius
R = (s/2)√(4 + 2√2)
Short Diagonal
d₁ = s√(2 + √2)
Medium Diagonal
d₂ = s(1 + √2)
Long Diagonal
d₃ = s√(4 + 2√2)
Interior Angle
°
Each interior angle
Octagon Formulas
Area
A = 2(1 + √2) × s²
Perimeter
P = 8s
Inradius (Apothem)
r = (s/2)(1 + √2)
Circumradius
R = (s/2)√(4 + 2√2)
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About Octagon Calculator
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 (135° each). It is a common shape found in everyday life—from stop signs to architectural designs.
How to Use This Calculator
From Side Length
- Select From Side Length mode
- Enter the side length of your octagon
- Instantly view all calculated properties: area, perimeter, inradius, circumradius, and diagonals
From Area
- Select From Area mode
- Enter the known area of the octagon
- The calculator determines the side length and all other properties
From Perimeter
- Select From Perimeter mode
- Enter the total perimeter
- View the calculated side length and all derived properties
Octagon Formulas
Core Formulas (Regular Octagon)
| Property | Formula | Description |
|---|---|---|
| Area | A = 2(1 + √2) × s² | Total area enclosed |
| Perimeter | P = 8s | Sum of all eight sides |
| Inradius (Apothem) | r = (s/2)(1 + √2) | Distance from center to middle of a side |
| Circumradius | R = (s/2)√(4 + 2√2) | Distance from center to a vertex |
Diagonal Formulas
| Property | Formula | Description |
|---|---|---|
| Short Diagonal | d₁ = s√(2 + √2) | Connects vertices with one vertex between |
| Medium Diagonal | d₂ = s(1 + √2) | Connects vertices with two vertices between |
| Long Diagonal | d₃ = s√(4 + 2√2) | Connects opposite vertices (passes through center) |
Derived Formulas
| Property | Formula |
|---|---|
| Side from Area | s = √(A / 2(1 + √2)) |
| Side from Perimeter | s = P / 8 |
Understanding Octagon Properties
Interior Angles
Every interior angle of a regular octagon measures exactly 135°. The sum of all interior angles is (8-2) × 180° = 1080°.
Inradius vs Circumradius
- Inradius (apothem): The perpendicular distance from the center to the midpoint of any side. It's the radius of the largest circle that fits inside the octagon.
- Circumradius: The distance from the center to any vertex. It's the radius of the smallest circle that completely contains the octagon.
Diagonals
A regular octagon has 20 diagonals total, falling into three categories:
- Short diagonals connecting vertices with one vertex between
- Medium diagonals connecting vertices with two vertices between
- Long diagonals connecting opposite vertices through the center
Real-World Applications
Architecture & Design
- Stop signs: The most iconic octagonal shape
- Buildings: Octagonal towers and rooms
- Windows: Decorative octagonal windows
Engineering
- Bolts and nuts: Some specialty hardware uses octagonal heads
- Floor tiles: Octagonal patterns in flooring
Nature & Science
- Crystals: Some mineral formations
- Molecular structures: Certain chemical compounds
Frequently Asked Questions
Why are stop signs octagonal?
The octagon shape was chosen for stop signs in 1923 because it was distinctive and could be recognized from any angle, even from behind. The unique 8-sided shape helps drivers identify it quickly.
What's the difference between a regular and irregular octagon?
A regular octagon has all sides equal and all angles equal (135°). An irregular octagon has sides and/or angles of different measures. This calculator works only for regular octagons.
How do I measure an octagon in real life?
Measure one side length, then use this calculator. For physical octagons, you can also measure the distance between opposite sides (which equals 2 × inradius) or opposite vertices (which equals the long diagonal).
How many triangles can be formed in an octagon?
A regular octagon can be divided into 8 isoceles triangles from the center, or into 6 triangles by drawing diagonals from one vertex.