Stop signs, gazebos, umbrellas and many floor tiles are octagons. A regular octagon is a square with its corners cut off at 45°, which is how carpenters build octagonal frames.
How to use the Octagon Calculator
- Side a
- Every result field is editable: type into Area or any other result and Side a plus every other field update to match.
Results update as you type, and everything is calculated in your browser.
How it works
Enter the side length or type into any result. The circumradius is half the long diagonal and the inradius half the medium diagonal (the width across flats).
What each result means
- Area = 2×(1 + sqrt(2))×[Side a]^2
- Perimeter = 8×[Side a]
- Long diagonal = [Side a]×sqrt(4 + 2×sqrt(2))
- Medium diagonal (width across flats) = [Side a]×(1 + sqrt(2))
- Short diagonal = [Side a]×sqrt(2 + sqrt(2))
- Circumradius = [Side a]×sqrt(4 + 2×sqrt(2))÷2
- Inradius = [Side a]×(1 + sqrt(2))÷2
Example values
How the main result changes when Side a changes and everything else stays at the example values above:
| Side a (cm) | Area (cm²) |
|---|---|
| 2.5 | 30.1777 |
| 3.75 | 67.8998 |
| 5 | 120.7107 |
| 6.25 | 188.6104 |
| 7.5 | 271.599 |
| 10 | 482.8427 |
Formula
A = 2(1 + √2)a², P = 8a
Long diagonal a√(4 + 2√2), medium a(1 + √2), short a√(2 + √2)
Worked example
Inputs — Side a: 5 cm.
Results — Area = 120.710678 cm²; Perimeter = 40 cm; Long diagonal = 13.06563 cm; Medium diagonal (width across flats) = 12.071068 cm; Short diagonal = 9.238795 cm; Circumradius = 6.532815 cm; Inradius = 6.035534 cm.
Key terms
- Across flats
- The distance between opposite parallel sides.
Frequently asked questions
What is the interior angle of an octagon?
135°.
How do I cut an octagon from a square?
Cut each corner off at 45°, removing a/√2 along each edge.
How many diagonals does an octagon have?
20.
How wide is an octagon across the flats?
a(1 + √2) ≈ 2.414a.