Question

The points of a six-pointed star consist of six identical equilateral triangles, with each side 4 cm (see figure). The area of the entire star, including the center, is $m\sqrt{3}$. Find the value of $m$?

Answer 1

You can think of this star as a large equilateral triangle with sides $12$ centimeters long, and three additional smaller equilateral triangles (shaded in the figure to the right) with sides 4 centimeters long.

Now the height of the larger equilateral triangle is $6\sqrt{3}$ and the height of the smaller equilateral triangle is $2\sqrt{3}$. Therefore, the areas of the triangles are as follows:

Large triangle: $A=\frac{b\times h}{2}=\frac{12\times 6\sqrt{3}}{2}=36\sqrt{3}$

Small triangles: $A=\frac{b\times h}{2}=\frac{4\times 2\sqrt{3}}{2}=4\sqrt{3}$

The total area of three smaller triangles and one large triangle is:

$36\sqrt{3}+3\left(4\sqrt{3}\right)=48\sqrt{3}c{m}^2$

Alternatively, you can apply the formula $A=\frac{s^2\sqrt{3}}{4}$

Small triangle: $A=\frac{4^2\sqrt{3}}{4}=\frac{16\sqrt{3}}{4}=4\sqrt{3}$

Next, add the area of the large triangle and the area of three smaller triangles, as above.