Hexagon Calculator

A regular hexagon consists of six equal sides and six equal interior angles of 120 degrees. Because it is geometrically symmetrical, every regular hexagon can be divided into six identical equilateral triangles radiating from its center. This unique property means that the circumradius (distance from the center to any vertex) is exactly equal to the side length ($s$). The foundational formulas driving this calculator are: Side ($s$), Perimeter ($P = 6s$), Apothem ($a = \frac{\sqrt{3}}{2}s$), Area ($A = \frac{3\sqrt{3}}{2}s^2$), Long Diagonal ($d_{long} = 2s$), and Short Diagonal ($d_{short} = \sqrt{3}s$). By entering any single variable, the tool solves for $s$ first and subsequently calculates all remaining geometric properties.

Let us walk through a manual calculation for a regular hexagon with a known side length of $s = 4$ units. First, find the perimeter by multiplying the side length by six: $P = 6 \times 4 = 24$ units. Next, determine the apothem using the formula $a = \frac{\sqrt{3}}{2} \times 4$, which simplifies to $2\sqrt{3} \approx 3.464$ units. To find the total area, apply the area equation: $A = \frac{3\sqrt{3}}{2} \times (4)^2 = \frac{3\sqrt{3}}{2} \times 16 = 24\sqrt{3} \approx 41.569$ square units. Finally, compute the diagonals: the long diagonal is twice the side length ($2 \times 4 = 8$ units), and the short diagonal is $\sqrt{3}$ times the side length ($4\sqrt{3} \approx 6.928$ units).

When solving geometry problems manually, always solve for the side length first, as it serves as the master key for all other measurements. Be mindful of rounding intermediate radical values like $\sqrt{3}$ too early, which can introduce compounding inaccuracies in large-scale architectural or manufacturing calculations. When dealing with irregular hexagons, remember that these specific formulas only apply to equilateral and equiangular regular hexagons.