A person makes cuts linking the midpoints of every 2 adjacent sides and distributes these 6 slices to his friends....

Question

A Greedy person is celebrating his birthday with 6 of his friends. His mother baked him a birthday cake in the shape of a regular hexagon. Wanting to keep most of the cake, he makes cuts linking the midpoints of every 2 adjacent sides, and distributes these 6 slices to his friends. What proportion of the cake does he have left for himself?

Answer 1

A regular hexagon is made of six congruent triangles of equal sides. Let L = side of the triangle. The area can be calculated as follows:
Area of initial hexagon = 6 * ( L * SQRT3 /2 L / 2 ) = 3/2 SQRT3* L^2
By linking the midpoints of two adjacent sides a smaller regular hexagon is formed of side equal to SQRT3/2 * L
Area of smaller hexagon = 3/2 SQRT3 * (SQRT3/2 * L) ^2 = 9/8 SQRT3 * L^2
By dividing the two areas, we obtain:
Area of smaller hexagon / area of initial hexagon = 9/8 SQRT3* L^2 / 3/2 SQRT3* L^2 = 3/4
The area of the remaining cake is 3/4 of the initial cake