Octagon Calculator
A regular octagon with a side length of 5 has an area of about 120.71 square units and a perimeter of 40 units. This octagon calculator finds the area, perimeter, apothem, and circumradius from a single input: the side length. Enter the side length and get all four measurements at once.
Quick answer
A regular octagon has eight equal sides and eight equal angles of 135 degrees.
Area
120.7107 sq units
Perimeter
40
Apothem
6.035533906
Circumradius
6.532814824
What this tells you
- •A regular octagon has eight equal sides and eight equal angles of 135 degrees.
- •The area formula is 2 times 1 plus the square root of 2, all multiplied by the side length squared.
- •The perimeter is simply 8 times the side length.
- •The apothem is the distance from the center to the midpoint of any side.
- •The circumradius is the distance from the center to any vertex.
How to Use
- 1Enter the side length of the regular octagon.
- 2Read the area, perimeter, apothem, and circumradius.
- 3All measurements use the same unit as the side length.
How It Works
Formula
area = 2(1 + √2) × s²The area of a regular octagon equals 2 times the quantity 1 plus the square root of 2, all multiplied by the side length squared. For a side of 5, this is 2 times 2.4142 times 25, which equals 120.71. The perimeter is 8 times the side. The apothem equals the side divided by 2 times the tangent of pi over 8. The circumradius equals the side divided by 2 times the sine of pi over 8.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
Side length 5
Side length 1
Octagon Measurements by Side Length
| Side | Area | Perimeter | Apothem | Circumradius |
|---|---|---|---|---|
| 3 | 43.46 | 24 | 3.62 | 3.92 |
| 5 | 120.71 | 40 | 6.04 | 6.53 |
| 8 | 309.02 | 64 | 9.66 | 10.45 |
Common mistakes
- Using the wrong area formula. The regular octagon formula has the 1 plus root 2 factor, not a simple polygon formula.
- Confusing apothem with circumradius. The apothem touches the midpoint of a side, while the circumradius reaches a vertex.
- Measuring side length along a diagonal or edge incorrectly.
Embed this calculator on your site
Drop this single line where you want the calculator to appear. It is responsive, mobile-friendly, resizes automatically, and is free to use with attribution.
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