Library / Geometry, coordinates, surveying and trigonometry / GEO-03
Geometry, coordinates, surveying and trigonometry
Apothem of a regular polygon
Distance from the centre of a regular polygon to the midpoint of a side, from the side length and the number of sides.
- Ref
- GEO-03
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- a [cm]
The apothem is the perpendicular distance from the centre of a regular polygon to the middle of a side. Joining the centre to the ends of one side makes an isosceles triangle with apex angle 2π/n. Half of it is a right triangle with angle π/n and opposite side s/2, so a = s/(2 tan(π/n)). It is the radius of the inscribed circle.
Distance from the centre of a regular polygon to the midpoint of a side, from the side length and the number of sides. The calculator uses half of the central angle, π/n in radians.
Calculator
- Circumradius, s/(2 sin(π/n))6 cm
- Area, n s a / 293.53 cm²
- Central angle, 360°/n60 °
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A regular polygon with 6 sides of 6 cm has an apothem of 5.196 cm, and a circumradius of 6 cm.
- A hexagon has an apothem of about 0.866 times its side, which is why hex bar sizes are quoted across the flats.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm) | Local step | Change in result |
|---|---|---|---|---|---|---|
| s Side length | 1 to 30 cm | 0.866 | 25.98 | 25.11 | +10% | +10 % |
| n Number of sides | 3 to 20 | 1.732 | 18.94 | 17.21 | +1 | +19.9 % |
Sweep of Number of sides
| Number of sides | Apothem (cm) |
|---|---|
| 3 | 1.732 |
| 5.125 | 4.265 |
| 7.25 | 6.484 |
| 9.375 | 8.615 |
| 11.5 | 10.71 |
| 13.63 | 12.78 |
| 15.75 | 14.84 |
| 17.88 | 16.89 |
| 20 | 18.94 |
Sweep one variable
Variables and how each one contributes
Apothem
What the equation gives you. Everything else feeds it.
Side length
The length of one side, in cm. A longer side gives a larger apothem in direct proportion. Raising it by 10 % (to 6.6) moves the result by +10 %, from 5.196 to 5.716 cm.
Number of sides
The number of sides, a whole number from 3. More sides push the apothem up towards the value of a circle of the same perimeter, but with diminishing returns. Raising it by 1 (to 7) moves the result by +19.9 %, from 5.196 to 6.23 cm.
Worked example with the values above
- Half the central angle is π/n = π/6 = 0.5236 rad.
- Tangent: tan(0.5236) = 0.57735.
- a = 6 / (2 × 0.57735) = 5.1962 cm.
Assumptions and limits
- A flat (plane) shape with straight sides, measured in one consistent unit.
- For regular polygons all sides and all angles are equal.
Common mistakes
- Using a sloping side as the height when it is not perpendicular to the base.
- Mixing units, such as a base in mm and a height in cm.
- Using the diagonal of one shape in the formula for another.
Statistics and uncertainty
These are exact formulas, so the result is only as good as the lengths measured. Area scales with the square of a length, so a 1 % error in a length gives about a 2 % error in the area.
Where it is used
- Plate, panel and gasket areas from simple drawings.
- Checking the size of regular flanges, hex bars and tiles.
Maths you need
- Area: the flat space inside a shape, in square units
- Perpendicular: at a right angle to
- Tangent of an angle: opposite over adjacent