# Spike in the Hive

Our next Catriona Shearer puzzle.
12.
Spike in the Hive

Two of the regular hexagons are identical; the third has area 10. What’s the area of the red triangle?

Spoiler!

First, what’s the area of the big hexagons?

There’s a little equilateral triangle in the middle between the hexagons, and its equilateralness means a big hex side is equal to a little hex side plus a triangle side ($H=h+t$, from the upper big hex), but also a big hex side is two little hex sides minus a triangle side ($H=2h-t$, from the lower big hex), and that means $t = h/2$ and $H = 3/2 h$. So the area of a big hex is $9/4\times 10$.

Now, the red triangle area: It’s half base times height, so it’s twice the area of a triangle with the same base and half the height, or twice the area of the same triangle sheared parallel to its base:

And that last triangle clearly is 1/3 the area of the hexagon it’s inscribed in. So $A = 2/3\times 9/4\times 10 = 15.$