Find the area of a regular hexagon with the given measurement 6-inch radius A equals sq in?

Answer 1

This is a complicated answer, so you should draw it as you read it. The formula for finding the area of any regular polygon is 1/2 times the perimeter times the apothem (a segment that goes from the center of the figure to the midpoint of a side. We have to know how long each side is to get the perimeter, and we need to know the apothem. Here's how we do that: A hexagon has 6 sides. Draw it. The formula for finding out the measurement for each angle inside this figure is [(n-2)*180]/n, where n is the number of sides. Since n is 6, the formula tells us that each interior angle is 120 degrees. Now draw a segment from each vertex (corner) of the figure to the center of the figure. Each of these is a radius, and each radius has cut each interior angle in half (60 degrees for each half). You should now have a figure that is divided into 6 congruent (equal) triangles. Let's look at just one of these triangles. Since it has two angles that are 60 degrees, the third angle must also be 60 degrees, since all three angles on any triangle add up to 180 degrees; therefore it is an equilateral triangle. That means that ALL its sides are 6 inches. Since one of these sides also a side of the hexagon, each of the sides of the hexagon is 6 inches, making its perimeter 36 inches. Now draw the apothem. This would be a segment that goes from the center of the hexagon to the midpoint of a side, or, on your one triangle, it goes from the top (vertex) of the triangle to the middle of the base of the triangle. Notice that this apothem has divided your triangle into two right triangles. You can figure out how long it is with trigonometry (sine) or by using a special triangle (30-60-90) comparison. Since the apothem divided the base of the triangle into two equal halves, the half-base is 3 inches long. Using the base angle of 60 degrees, we know that sin 60 degrees is the opposite (the apothem)/hypotenuse, which is x/6. The sin 60 = .8660, so .8660 = x/6. x = 5.196. Plug this in for the apothem in the formula, 1/2 P*a, and you get 1/2 * 36 * 5.196, which is 93.5 square inches.