Thomas is making a sign in the shape of a regular hexagon with 4-inch sides, which he will cut out from a rectangular sheet of metal. What is the sum of the areas of the four triangles that will be removed from the rectangle?

So it’ll look like this:

It’s helpful just to know that a regular hexagon’s interior angles all measure 120°, but you can also calculate that using :

That means that the four triangles you’re cutting off the rectangle are each 30°-60°-90° triangles with 4-inch hypotenuses.

Those will have legs of 2 and , and therefore areas of . Since there are four such rectangles, the total area you’re cutting off is

Daniel Sedric says:

how could you know that we’re dealing with 30-60-90 triangles just because each angle of the hexagon is 120 degrees? please elaborate on this.

It has to do with supplementary angles. The cuts are being made to a side of a rectangle, so it starts as a straight, 180° surface. To make a cut and leave 120° left, you must cut out 60°. Does that help?