Octagon Calculator
Instantly calculate the area, perimeter, diagonals, inradius, and circumradius of any regular octagon. Enter one known measurement and get every value in a click.
Octagon Measurements
Enter any one known value and select its type — all other properties will be calculated automatically
Choose which measurement you already know.
Enter the length of one side of the octagon.
Octagon Properties
Area, perimeter, diagonals, and circle radii
Enter a measurement above and click Calculate Octagon to see the full set of octagon properties.
Octagon Formulas
All the mathematical formulas used by this calculator, expressed in terms of the side length a.
Area
≈ 4.82843 × a²
The area enclosed by all eight sides. Scales with the square of the side length.
Perimeter
The total boundary length — simply the side length multiplied by the eight equal sides.
Circumradius (R)
≈ 1.30656 × a
The radius of the circle passing through all 8 vertices of the octagon.
Inradius (r)
≈ 1.20711 × a
The radius of the largest circle that fits inside the octagon, tangent to each side.
Long Diagonal (d)
≈ 2.61313 × a
The longest diagonal, connecting two directly opposite vertices. Equal to 2R.
Interior Angles
Total sum = 1,080°
All interior angles of a regular octagon are equal. The sum is (8−2) × 180° = 1,080°.
Octagon Properties by Side Length
Common side lengths and their corresponding octagon measurements for fast reference. All values are rounded to two decimal places.
| Side (a) | Area (A) | Perimeter (P) | Circumradius (R) | Inradius (r) | Long Diagonal (d) |
|---|---|---|---|---|---|
| 1 | 4.83 | 8.00 | 1.31 | 1.21 | 2.61 |
| 2 | 19.31 | 16.00 | 2.61 | 2.41 | 5.23 |
| 3 | 43.46 | 24.00 | 3.92 | 3.62 | 7.84 |
| 5 | 120.71 | 40.00 | 6.53 | 6.04 | 13.07 |
| 8 | 309.02 | 64.00 | 10.45 | 9.66 | 20.91 |
| 10 | 482.84 | 80.00 | 13.07 | 12.07 | 26.13 |
| 12 | 694.90 | 96.00 | 15.68 | 14.49 | 31.36 |
| 15 | 1086.40 | 120.00 | 19.60 | 18.11 | 39.20 |
| 20 | 1931.37 | 160.00 | 26.13 | 24.14 | 52.26 |
| 25 | 3017.77 | 200.00 | 32.66 | 30.18 | 65.33 |
| 50 | 12071.07 | 400.00 | 65.33 | 60.36 | 130.65 |
| 100 | 48284.27 | 800.00 | 130.65 | 120.71 | 261.31 |
Octagon Calculator FAQ
Answers to the most frequently asked questions about octagon geometry, formulas, and real-world applications.
A regular octagon is an eight-sided polygon where all sides are equal in length and all interior angles are equal. Each interior angle measures exactly 135 degrees, and the sum of all interior angles is 1,080 degrees. The most recognisable real-world example is the standard stop sign.
The area of a regular octagon with side length a is calculated using the formula: Area = 2(1 + √2) × a². For example, if a side measures 5 cm, the area is approximately 2 × 2.41421 × 25 = 120.71 cm². This calculator handles the formula for you — just enter the side length and click Calculate.
The perimeter of a regular octagon is simply 8 times the side length. If each side is a, then Perimeter = 8a. For example, a regular octagon with a 5 cm side has a perimeter of 40 cm. Because all sides are equal, this is the most straightforward of the octagon properties to calculate.
A regular octagon has 20 diagonals in total, calculated using the formula n(n−3)/2 = 8×5/2 = 20. These fall into three distinct groups: short diagonals connecting vertices two positions apart, medium diagonals connecting vertices three positions apart, and long diagonals connecting directly opposite vertices, of which there are exactly 4.
The circumradius (R) is the radius of the circle passing through all 8 vertices of the octagon. The inradius (r) is the radius of the largest circle that can fit inside the octagon, touching each side at its midpoint (also called the apothem). For a side of length a: R ≈ 1.30656a and r ≈ 1.20711a. The circumradius is always slightly larger than the inradius.
Stop signs are the most common everyday example of a regular octagon. Other examples include some bolts and nuts (though most are hexagonal), bathroom and kitchen floor tiles, certain clock faces, decorative gazebos, and the famous UFC Octagon ring used in mixed martial arts competitions. Octagonal designs also appear frequently in architecture, particularly in church towers, windows, and floor plans.