What is the hexagon’s area?
Type any one measure of a regular hexagon, such as its side, area or long diagonal. The hexagon calculator finds the side and then every other measure from it.
- Area (A)
- 41.5692 in²
A regular hexagon with side 4 in has an area of 41.5692 in² and a perimeter of 24 in.
- Perimeter (P)
- 24 in
- Apothem (inradius)
- 3.464 in
- Long diagonal (D)
- 8 in
- Short diagonal (d)
- 6.928 in
Area (A): 41.5692 in². A regular hexagon with side 4 in has an area of 41.5692 in² and a perimeter of 24 in.
How to calculate
Finds every measure of a regular hexagon from any one of them: side, perimeter, area, apothem, long diagonal and short diagonal.
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.
Formula: P = 6s; apothem = s√3 ÷ 2; A = ½ × apothem × P = (3√3 ÷ 2)s²; long diagonal D = 2s; short diagonal d = s√3.
- 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
Each example is checked against the calculator on every build.
- Side (s) 157.5 in gives Perimeter (P) 944.9 in, Apothem (inradius) 136.4 in, Area (A) 64,432.4 in², Long diagonal (D) 315 in, Short diagonal (d) 272.8 in.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
- Perimeter (P) 1,181 in gives Side (s) 196.9 in, Area (A) 100,676 in², Apothem (inradius) 170.5 in.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
- Area (A) 155,000 in² gives Side (s) 244.3 in, Perimeter (P) 1,466 in.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
- Long diagonal (D) 393.7 in gives Side (s) 196.9 in, Short diagonal (d) 341 in.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
How it works
A regular hexagon has six equal sides s. Lines from the center to the corners cut it into six equilateral triangles with side s. From that:
- Perimeter: P = 6s.
- Apothem (center to the middle of a side): ap = s√3 ÷ 2, the height of one equilateral triangle.
- Area: A = ½ × ap × P = (3√3 ÷ 2)s².
- Long diagonal (opposite corners, through the center): D = 2s.
- Short diagonal (two corners with one between them): d = s√3.
Type one measure. The page finds the side from it (s = P ÷ 6, s = 2ap ÷ √3, s = √(2A ÷ (3√3)), s = D ÷ 2, or s = d ÷ √3), then every other measure from the side. Two typed measures that do not fit one hexagon give no answer.
Rules
- Every measure must be more than 0.
- The side can be at most 10⁹ m, the perimeter 6 × 10⁹ m, the area 3 × 10¹⁸ m², the apothem 10⁹ m, and each diagonal 2 × 10⁹ m. A value worked out past its limit gives no answer.
Output format. Values are double-precision numbers in the unit shown next to each field. √3 is the double-precision square root of 3.
Sources are OpenStax Contemporary Mathematics, because Algebra and Trigonometry 2e has no section on regular polygons.
Worked examples by hand
Side 4. P = 6 × 4 = 24. ap = 4√3 ÷ 2 = 2√3 = 3.4641. A = ½ × 2√3 × 24 = 24√3 = 41.5692. D = 8. d = 4√3 = 6.9282.
Perimeter 30. s = 30 ÷ 6 = 5. A = (3√3 ÷ 2) × 25 = 64.9519. ap = 5√3 ÷ 2 = 4.3301.
Area 100. s = √(200 ÷ (3√3)) = √38.4900 = 6.2040. P = 6 × 6.2040 = 37.2242.
Long diagonal 10. s = 10 ÷ 2 = 5. d = 5√3 = 8.6603.
Other questions people ask
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.