Question

Two similar regular hexagons have a common center. If each side of the big hexagon is twice the side of the small one and the area of the small hexagon is 3 sq. in, what is the area of the big hexagon?

Answer 1

The area of the larger square is 4 times larger than the area of the smaller square. The area of the big hexagon is 12 sq. in.

Step-by-step explanation:

The area of the larger square is 4 times larger than the area of the smaller square. This is because the area of a square is proportional to the square of its side length.

In this case, the side length of the larger square is twice the side length of the smaller square, so the area of the larger square is 2² times greater than the area of the smaller square.

Given that the area of the small hexagon is 3 sq. in, the area of the big hexagon can be found by multiplying the area of the small hexagon by the square of the scale factor:

Area of big hexagon = (scale factor)² * Area of small hexagon = 2² * 3 sq. in = 12 sq. in

Answer 2

To determine any area of a shape we need to know the formula for the area of that particular shape which is for a regular hexagon is

Area of hexagon = (3 √3) a^2 / 2
where a is the length of the side of the hexagon
So, first, we determine the length of the side of the smaller hexagon from the given area.

3 in^2 = (3 √3) a^2 / 2
6 in^2 = (3 √3) a^2
a^2 = 2 √3 / 3
a = 1.0746 in

The length of the side of the bigger hexagon is twice that of the smaller so,
A = 2a = 2 (1.0746 in ) = 2.1491 in

Area of the bigger hexagon = (3 √3) a^2 / 2
= (3 √3) 2.1491^2 / 2 = 12 in^2