An equilateral triangle and a regular hexagon have perimeters of the same length. If the triangle’s area is 2 square units, what is the area of the hexagon? Unless you’re a budding John Von Neuman, it isn’t necessary to calculate areas and then divide. That’s the hard way.

Suppose the triangle’s perimeter is 12. Then each of its sides is 4, and each side of the hexagon is 2.

Partition the two shapes into congruent equilateral triangles.

There are four in the big triangle and six in the hexagon. Since the big triangle’s area is 2, each small triangle has area ½. So the hexagon’s area is 6 x ½ = 3.