Vexing Hexes

[Garnfellow]

It's been a long time since high school geometry, but I'm not sure about T5's definitions of hexes.

"Hex size (or hex diameter) reflects the distance from the center of a hex to the center of an adjacent similarly sized hex" (37). A World Hex is 1,000 km, a Terrain Hex is 100 km, and a Local Hex is 10 km.

Again and again, Book 3 refers to the distance across a hex as the diameter. For example, "Each Single Hex is 1 km in diameter (from the center of the Single Hex to the center of an adjacent Single Hex)" (41).

So far so good. Book 1 describes a Terrain Hex as "6,500 square km" and a Local Hex as "approximately 65 square km" (41).

But here's a diagram of hexagon

a is the length of the side, R is the circumcircle radius, d is the long diagonal (2 R), r is the apothem, and S is the short diagonal (2 r). The distance from the center of a hex to the center of an adjacent hex is not 2 R, but 2 x apothem (2 r), or S, the short diagonal.

The area of a regular hexagon is 6 x apothem squared / square root of 3. The area of a Local Hex, then, should be 6 x (10/2)^2 / square root of 3, or 87 square km. The area of a Terrain Hex 8,660 (call it 8,700 sq km), and the area of a World Hex 870,000 km.

I believe this only affects the areas in Book 1 and not anything in Book 3. Or have I completely lost it?

 

[Garnfellow]

tl;dr version: the area of a 10 km Local Hex should be 87 sq km, instead of the 65 sq km given in Book 1. This is approximately 8,700 hectares or 21,500 acres. The area of a 100 km Terrain Hex should be 8,700 sq km instead of 6,500 sq km. The area of a 1,000 km World Hex is approximately 870,000 sq km. And the area of a 1 km Single Hex is approximately 1 sq km.

 

[Original_Carl]

Hexagon Calculator

This site helps me a lot with the math stuff on Hexes.

FWIW the area of a Single Hex (1km vertex to vertex) is .65km according to Pythagoras.

 

[Garnfellow]

Hexagon Calculator

This site helps me a lot with the math stuff on Hexes.

FWIW the area of a Single Hex (1km vertex to vertex) is .65km according to Pythagoras.

Yup, and that's likely why T5 gives 65 sq km as the area of a Local Hex. Trouble is, the length of vertex to vertex isn't equal to the distance between the center of a hex and the center of an adjacent hex.