Hello, mathaddict!
I made a sketch . . . and the answer jumped out at me!
An equilateral triangle is inscribed in a regular hexagon.
This triangle's vertices touch the midpoints of three sides of the regular hexagon.
The area of the triangle is 12 cm².
What is the area of the regular hexagon in cm² ?
The diagram looks like this: Code:
*---*---*
/ / \ \
/ / \ \
/ / \ \
* / \ *
\ / \ /
*-----------*
\ /
*-------*
Divide the hexagon into "unit triangles",
. . and we have this diagram: Code:
*---*---*
/ \ /x\ / \
*---*---*---*
/ \ /x\x/x\ / \
*---*---*---*---*
\ /x\x/x\x/x\ /
*---*---*---*
\ / \ / \ /
*---*---*
We see that the triangle occupies of the hexagon.
Therefore, the hexagon has: .