Area of Octagon Calculator
Enter the length of one side of a regular octagon and choose a unit to get its area in the matching square unit. The guide below shows the formula, where it comes from and how to use it.
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Created by the Reeliy Editorial Team. Last updated: Oct 19, 2026
Area of an octagon formula
A regular octagon has eight equal sides and eight equal angles of 135°. If the side length is a, its area is
A = 2 × (1 + √2) × a² ≈ 4.828427 × a²
The constant comes from a simple picture. Draw the smallest square that holds the octagon, so that its sides touch the octagon's flat edges. That square is a × (1 + √2) wide, because each corner you trim away is an isosceles right triangle whose short sides are a ÷ √2 long. Each of the four trimmed corners has area a² ÷ 4, so together they remove a² from the square.
A = a² × (1 + √2)² − a² = a² × (3 + 2√2 − 1) = a² × (2 + 2√2)(1 + √2)² = 3 + 2√2, which gives the same 2 × (1 + √2) as above.
How to use the calculator
- Type the length of one side into Octagon Side. Zero or a positive number is accepted.
- Pick the unit of that length from the drop-down: millimetres, centimetres, decimetres, metres, kilometres, inches, feet, yards or miles.
- Press Calculate. The area is shown in the square of the same unit, for example cm² when the side is in cm.
The answer is rounded to six decimal places. Trailing zeros are trimmed in pairs, so a whole number prints bare and thousands are separated with commas.
The calculator does not convert between units. A side of 5 cm gives an area in cm², not in m². To change the unit of the area, change the unit of the side before you calculate.
Examples: find the area of an octagon
Example 1: a tile. A regular octagonal tile has sides of 5 cm. The area is 4.828427 × 5² = 4.828427 × 25 = 120.710678 cm².
Example 2: a garden pavilion. The floor of an octagonal pavilion has sides of 2.4 m. The area is 4.828427 × 2.4² = 4.828427 × 5.76 = 27.811740 m², which is the amount of decking or flooring needed before any cutting waste.
Example 3: you only know the width across the flats. Suppose a panel measures 30 cm from one flat side to the opposite one. The side length is 30 ÷ (1 + √2) ≈ 12.426407 cm. Put that in the calculator and you get 745.584412 cm². As a shortcut, the area equals 2 × (√2 − 1) × d² ≈ 0.828427 × d², where d is the width across the flats: 0.828427 × 900 = 745.58 cm².
Other measurements of a regular octagon
Everything about a regular octagon scales with its side a. The table lists the common ones, with the values for a side of 5 cm in the last column.
| Measurement | Formula | Factor × a | For a = 5 cm |
|---|---|---|---|
| Perimeter | 8a | 8 | 40 cm |
| Width across the flats | a × (1 + √2) | 2.414214 | 12.071068 cm |
| Apothem (centre to a flat side) | a × (1 + √2) ÷ 2 | 1.207107 | 6.035534 cm |
| Circumradius (centre to a corner) | a ÷ (2 × sin 22.5°) | 1.306563 | 6.532815 cm |
| Width across the corners | a ÷ sin 22.5° | 2.613126 | 13.065630 cm |
| Area | 2 × (1 + √2) × a² | 4.828427 × a | 120.710678 cm² |
Another way to reach the area is half the perimeter times the apothem: ½ × 40 × 6.035534 = 120.710678 cm², the same answer as before. For the perimeter of an octagon and other polygons, use the perimeter calculator.
Areas for common side lengths
| Side | Area |
|---|---|
| 1 | 4.828427 square units |
| 2 | 19.313708 square units |
| 3 | 43.455844 square units |
| 5 | 120.710678 square units |
| 10 | 482.842712 square units |
| 12 | 695.293506 square units |
Because the area depends on the square of the side, doubling the side makes the area four times larger: 19.313708 is four times 4.828427.
Limits and assumptions
- The formula is for a regular octagon only. If the sides or angles differ, split the shape into rectangles and triangles and add their areas, for example with the area of a rectangle and area of a triangle calculators.
- Measure the side of the finished shape, not the width of the bounding square.
- Plain decimal numbers are expected. Entries such as 1e2 or .5 are rejected, so write 100 or 0.5 instead.
Frequently asked questions
What is the formula for the area of a regular octagon?
A = 2 × (1 + √2) × a², where a is the side length. That is about 4.828427 times the side squared.
How do I find the area if I only know the width across the flats?
Multiply the square of that width by 2 × (√2 − 1), about 0.828427. For a width of 30 cm the area is 0.828427 × 900 ≈ 745.58 cm².
Why does the formula contain the square root of 2?
An octagon is a square with its four corners cut off. Each cut corner is an isosceles right triangle, and the diagonal geometry of such a triangle introduces √2.
In what unit is the answer?
In the square of the unit you picked for the side: cm² for centimetres, in² for inches, mile² for miles, and so on. The calculator does not convert between units.
Does the calculator work for an irregular octagon?
No. It assumes eight equal sides and equal angles. For an irregular octagon, divide it into simpler shapes and add up their areas.
Cite this calculator and page
If you use this page in your work, you can cite it as:
Reeliy Editorial Team. (2026). Area of Octagon Calculator. Reeliy. https://reeliy.com/calculators/octagon-area-calculator“Area of Octagon Calculator.” Reeliy, 2026, https://reeliy.com/calculators/octagon-area-calculator.