# What is the hexagon’s area?

Finds every measure of a regular hexagon from any one of them: side, perimeter, area, apothem, long diagonal and short diagonal.

- Page: https://www.acalculator.org/math/hexagon-calculator
- JSON spec: https://www.acalculator.org/math/hexagon-calculator.json
- Version: 11b28589c725

## Default answer

Example with the default inputs (Side (s) 4 in): A regular hexagon with side 4 in has an area of 41.5692 in² and a perimeter of 24 in.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| s | Side (s) | The length of one side of the regular hexagon. |
| p | Perimeter (P) | The distance around the hexagon: six sides. |
| a | Area (A) | The space inside the hexagon: 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. |
| d | Long diagonal (D) | The distance between opposite corners, through the center: two sides. |
| ds | Short diagonal (d) | The distance between two corners with one corner between them: the side times √3. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| s | Side (s) | The length of one side of the regular hexagon. |
| p | Perimeter (P) | The distance around the hexagon: six sides. |
| a | Area (A) | The space inside the hexagon: 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. |
| d | Long diagonal (D) | The distance between opposite corners, through the center: two sides. |
| ds | Short diagonal (d) | The distance between two corners with one corner between them: the side times √3. |

## Method

P = 6s; apothem = s√3 ÷ 2; A = ½ × apothem × P = (3√3 ÷ 2)s²; long diagonal D = 2s; short diagonal d = s√3.

## Assumptions

- The hexagon is regular: six equal sides and six equal angles of 120°.
- The radius (center to a corner) equals the side, so the long diagonal is two sides.
- Type one measure; the side follows from it, and every other measure from the side.

## Worked examples

1. s = 4 gives p = 24, ap = 3.464102, a = 41.569219, d = 8, ds = 6.928203. Source: OpenStax, Contemporary Mathematics, §10.6 Area (regular polygon A = ½ap; a regular hexagon with 4 cm sides and apothem 2√3 has area 24√3 ≈ 41.57 cm²). https://openstax.org/books/contemporary-mathematics/pages/10-6-area.
2. p = 30 gives s = 5, a = 64.951905, ap = 4.330127. Source: OpenStax, Contemporary Mathematics, §10.4 Polygons, Perimeter, and Circumference (P = n·s; each angle of a regular hexagon is 120°). https://openstax.org/books/contemporary-mathematics/pages/10-4-polygons-perimeter-and-circumference; OpenStax, Contemporary Mathematics, §10.6 Area (regular polygon A = ½ap; a regular hexagon with 4 cm sides and apothem 2√3 has area 24√3 ≈ 41.57 cm²). https://openstax.org/books/contemporary-mathematics/pages/10-6-area.
3. a = 100 gives s = 6.204032, p = 37.224194. Source: OpenStax, Contemporary Mathematics, §10.6 Area (regular polygon A = ½ap; a regular hexagon with 4 cm sides and apothem 2√3 has area 24√3 ≈ 41.57 cm²). https://openstax.org/books/contemporary-mathematics/pages/10-6-area.
4. d = 10 gives s = 5, ds = 8.660254. Source: OpenStax, Contemporary Mathematics, §10.4 Polygons, Perimeter, and Circumference (P = n·s; each angle of a regular hexagon is 120°). https://openstax.org/books/contemporary-mathematics/pages/10-4-polygons-perimeter-and-circumference.

## FAQ

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

Use A = (3√3 ÷ 2)s², or half the apothem times the perimeter. A hexagon with 4 cm sides has apothem 2√3 cm and perimeter 24 cm, so A = ½ × 2√3 × 24 = 24√3 ≈ 41.57 cm².

### What is the apothem of a hexagon?

The distance from the center to the middle of a side, at a right angle to the side. In a regular hexagon it is s√3 ÷ 2, about 0.866 times the side.

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

Rearrange A = (3√3 ÷ 2)s²: s = √(2A ÷ (3√3)). An area of 100 in² gives a side of about 6.204 in.

### What are the long and short diagonals?

The long diagonal joins opposite corners through the center and is 2s. The short diagonal skips one corner and is s√3. A hexagon with 5 in sides has diagonals of 10 in and about 8.660 in.

### What is the angle inside a regular hexagon?

Each inside angle is 120°. The angles of a six-sided polygon add up to (6 − 2) × 180° = 720°, and 720° ÷ 6 = 120°.

### Why is the distance from the center to a corner equal to the side?

Lines from the center to the six corners cut a regular hexagon into six equilateral triangles. Each triangle has all three sides equal to the hexagon’s side.

## Sources

- OpenStax, Contemporary Mathematics, §10.6 Area (A = ½ap for a regular polygon; a regular hexagon with 4 cm sides and apothem 2√3 has area 24√3 ≈ 41.57 cm²). https://openstax.org/books/contemporary-mathematics/pages/10-6-area (retrieved 2026-10-02)
- OpenStax, Contemporary Mathematics, §10.4 Polygons, Perimeter, and Circumference (P = n·s; a regular hexagon’s angles add to 720°, 120° each). https://openstax.org/books/contemporary-mathematics/pages/10-4-polygons-perimeter-and-circumference (retrieved 2026-10-02)