# What is the octagon’s area?

Finds every measure of a regular octagon from any one of them: side, perimeter, area, apothem, circumradius, width and the long and short diagonals.

- Page: https://www.acalculator.org/math/octagon-calculator
- JSON spec: https://www.acalculator.org/math/octagon-calculator.json
- Version: bfd81d1d794b

## Default answer

Example with the default inputs (Side (s) 5 in): A regular octagon with side 5 in has an area of 120.711 in² and a perimeter of 40 in.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| s | Side (s) | The length of one side of the regular octagon. |
| p | Perimeter (P) | The distance around the octagon: eight sides. |
| a | Area (A) | The space inside the octagon: half the apothem times the perimeter. |
| ap | Apothem (inradius) | The distance from the center to the middle of a side, at a right angle to the side. |
| r | Circumradius (R) | The distance from the center to a corner. |
| w | Width across flats | The distance between two opposite sides: twice the apothem, also the middle diagonal. |
| d | Long diagonal (D) | The distance between opposite corners, through the center. |
| ds | Short diagonal | The distance between two corners with one corner between them. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| s | Side (s) | The length of one side of the regular octagon. |
| p | Perimeter (P) | The distance around the octagon: eight sides. |
| a | Area (A) | The space inside the octagon: half the apothem times the perimeter. |
| ap | Apothem (inradius) | The distance from the center to the middle of a side, at a right angle to the side. |
| r | Circumradius (R) | The distance from the center to a corner. |
| w | Width across flats | The distance between two opposite sides: twice the apothem, also the middle diagonal. |
| d | Long diagonal (D) | The distance between opposite corners, through the center. |
| ds | Short diagonal | The distance between two corners with one corner between them. |

## Method

P = 8s; apothem = s(1 + √2) ÷ 2; A = ½ × apothem × P = 2(1 + √2)s²; R = s√(4 + 2√2) ÷ 2; width = s(1 + √2); long diagonal = s√(4 + 2√2); short diagonal = s√(2 + √2).

## Assumptions

- The octagon is regular: eight equal sides and eight equal angles of 135°.
- Type one measure; the side follows from it, and every other measure from the side.

## Worked examples

1. s = 5 gives p = 40, ap = 6.035534, a = 120.710678, r = 6.532815, w = 12.071068, d = 13.06563, ds = 9.238795. Source: OpenStax, Contemporary Mathematics, §10.6 Area (regular polygon A = ½ap, with a the apothem and p the perimeter). https://openstax.org/books/contemporary-mathematics/pages/10-6-area.
2. p = 16 gives s = 2, a = 19.313708, w = 4.828427. Source: OpenStax, Contemporary Mathematics, §10.4 Polygons, Perimeter, and Circumference (P = n·s; the angles of an n-gon add to (n − 2) × 180°). https://openstax.org/books/contemporary-mathematics/pages/10-4-polygons-perimeter-and-circumference; OpenStax, Contemporary Mathematics, §10.6 Area (regular polygon A = ½ap, with a the apothem and p the perimeter). https://openstax.org/books/contemporary-mathematics/pages/10-6-area.
3. a = 100 gives s = 4.550899, p = 36.407189. Source: OpenStax, Contemporary Mathematics, §10.6 Area (regular polygon A = ½ap, with a the apothem and p the perimeter). https://openstax.org/books/contemporary-mathematics/pages/10-6-area.
4. w = 10 gives s = 4.142136, ap = 5. Source: OpenStax, Contemporary Mathematics, §10.4 Polygons, Perimeter, and Circumference (P = n·s; the angles of an n-gon add to (n − 2) × 180°). https://openstax.org/books/contemporary-mathematics/pages/10-4-polygons-perimeter-and-circumference.

## FAQ

### How do I find the area of a regular octagon?

Use A = 2(1 + √2)s², or half the apothem times the perimeter. An octagon with 5 cm sides has area 2(1 + √2) × 25 = 50(1 + √2) ≈ 120.71 cm².

### What is the apothem of an octagon?

The distance from the center to the middle of a side, at a right angle to the side. In a regular octagon it is s(1 + √2) ÷ 2, about 1.2071 times the side.

### How do I find the side from the width of an octagon?

The width across flats (between two opposite sides) is s(1 + √2), so s = width ÷ (1 + √2) = width × (√2 − 1). A 10 in wide octagon, such as a stop sign shape, has sides of about 4.142 in.

### How do I find the side from the area?

Rearrange A = 2(1 + √2)s²: s = √(A ÷ (2(1 + √2))). An area of 100 in² gives a side of about 4.551 in.

### What are the diagonals of a regular octagon?

There are three lengths. The short diagonal skips one corner and is s√(2 + √2). The middle one joins corners three apart and equals the width, s(1 + √2). The long diagonal joins opposite corners through the center and is s√(4 + 2√2), twice the circumradius.

### What is each angle inside a regular octagon?

Each inside angle is 135°. The angles of an eight-sided polygon add up to (8 − 2) × 180° = 1,080°, and 1,080° ÷ 8 = 135°.

## Sources

- OpenStax, Contemporary Mathematics, §10.6 Area (A = ½ap for a regular polygon, with a the apothem and p the perimeter). https://openstax.org/books/contemporary-mathematics/pages/10-6-area (retrieved 2026-10-05)
- OpenStax, Contemporary Mathematics, §10.4 Polygons, Perimeter, and Circumference (P = n·s; the angles of an n-sided polygon add to (n − 2) × 180°). https://openstax.org/books/contemporary-mathematics/pages/10-4-polygons-perimeter-and-circumference (retrieved 2026-10-05)