The regular hexagon

A hexagon whose sides and interior angles are all equal is a regular hexagon¹. We examine the one centered at the origin with circumradius $4$ . Its vertices are $(4, 0)$ , $(2, 2\sqrt{3})$ , $(-2, 2\sqrt{3})$ , $(-4, 0)$ , $(-2, -2\sqrt{3})$ and $(2, -2\sqrt{3})$ .

Interior angles

The interior angles sum to $(6 - 2) \times 180^\circ = 720^\circ$ , so each one is $120^\circ$ .

It splits into six equilateral triangles

Joining the center to the six vertices cuts the hexagon into six triangles. The angles around the center are $\dfrac{360^\circ}{6} = 60^\circ$ each, and the other two sides are both radii, so every one of those triangles is equilateral.

The side length therefore equals the circumradius, here $4$ . Computing the distance between two adjacent vertices confirms it: $\sqrt{(4 - 2)^2 + (0 - 2\sqrt{3})^2} = \sqrt{16} = 4$ .

Area

An equilateral triangle of side $a$ has area $\dfrac{\sqrt{3}}{4}a^2$ , and there are six of them.

S = 6 \times \frac{\sqrt{3}}{4}a^2 = \frac{3\sqrt{3}}{2}a^2 = \frac{3\sqrt{3}}{2} \times 16 = 24\sqrt{3}

That is about $41.57$ .

The main quantities

Quantity	Formula	Value
Side	$a = R$	$4$
Inradius	$\dfrac{\sqrt{3}}{2}a$	$2\sqrt{3} \approx 3.46$
Perimeter	$6a$	$24$
Area	$\dfrac{3\sqrt{3}}{2}a^2$	$24\sqrt{3} \approx 41.57$

The distance from the center to a side is $2\sqrt{3}$ , the height of one of the equilateral triangles. That is why the top and bottom sides lie on the horizontal lines $y = \pm 2\sqrt{3}$ . The same distance is the radius of the inscribed circle.

Perimeter against circumference

The circumference of the circumscribed circle is $2\pi \times 4 = 25.13$ , so the perimeter of the hexagon is about $95$ percent of it. Approximating $\pi$ from the perimeter of a regular polygon is the idea of closing that gap by taking more and more sides.

Diagonals

How they join	Example	Length
Skipping one vertex	$(4, 0)$ and $(-2, 2\sqrt{3})$	$4\sqrt{3}$
Opposite vertices	$(4, 0)$ and $(-4, 0)$	$8$

The latter is the diameter of the circumscribed circle.

Tiling the plane

Along with the equilateral triangle and the square, the regular hexagon tiles the plane without gaps. Of those three it needs the shortest perimeter to enclose a given area, which is why a honeycomb takes this shape².

The six lines on the graph are the six sides, the two arcs are the circumscribed circle, and the large dots are the six vertices together with the center.

Hexagon, Wikipedia
Honeycomb conjecture, Wikipedia