Problem number six states. Six congruent rhombuses, each of area five centimeters square, form a star. The tips of the star are joined to form a regular hexagon as shown. What is the area of the hexagon? Let's start out with these six congruent rhombuses in a hexagon. The question asks us to find the area of this hexagon. The area of it consists of the inner six rhombuses on the remaining six triangular sections. We know the area of the rhombuses to be five, so we can say that area is six times five, or 30. Now let's figure out the area of the triangle. Let's start out with the central angle. Since all the rhombuses are congruent, we know that their inner angles are the same. We'll call this angle alpha. We know the sum of these angles is equal to a full rotation, so 360 degrees. Simplifying, we get six alpha equals 360. Dividing by six, we get alpha equals 60 degrees. We also know the other angle is also 60 degrees, as opposite angles on a rhombus are congruent. Next let's consider this angle. We'll call it beta. Beta is the inner angle of the regular hexagon, and the formula for finding this angle is the number of sides of the regular polygon minus two times 180 degrees, divided by the number of sides of the regular polygon. Replacing n with six, since our regular polygon is a hexagon, we get four times 180 over six. So beta is going to be equal to 120 degrees. Now let's consider one of the interior angles of the small triangle, gamma. We know that two of these in the interior angle of the rhombus form the inner angle of the hexagon, so we know two gamma plus 60 equals 120 degrees. Solving for this, we get that gamma is equal to 30 degrees. Now looking at our diagram, we might notice that the inner angles of the triangle are half of the inner angles of the rhombus, which might suggest the triangle is half of the rhombus. Let's try to prove this. Let's call the obtuse angle in the rhombus delta, and the obtuse angle in the triangle epsilon. Knowing that the sum of the angles of a rhombus equals 360 degrees, we can say that two of the acute angles plus two of the obtuse angles that compose the rhombus must sum to 360 degrees. Solving, we get that delta is equal to 120 degrees. Now let's solve for epsilon. Here we have the same story except we sum the angles of the triangle, and we know the angles of a triangle must sum up to 180 degrees. Solving, we get epsilon is equal to 120 degrees. So epsilon and delta are the same angle. This confirms the thoughts that we had earlier that the triangle is half the rhombus. If we split the rhombus on its long diagonal, we can see. So the rhombus had an area of 5. If we split in half, then each part would have an area of 2.5. And now we know that the triangle is just half the rhombus, so we know that its area is also 2.5. So back to our original equation. We know that the area of the triangle is 2.5, 6 times 2.5 is 15, and 30 plus 15 is 45, which means the area of our hexagon is 45. So the question asked us, what is the area of the hexagon? The answer is 45 centimeters squared. Letter C.

hexagon

rhombuses

area calculation

triangles

geometry