Q:

# Find the area of the hexagon whose apothem is 5 v3. Round to the nearest tenth if necessary.

Accepted Solution

A:
To find the area of a regular hexagon, we need to know its apothem and side length. The apothem is the distance from the center of the hexagon to the midpoint of a side, and the side length is the distance between two opposite vertices. Since the hexagon is regular, all sides and angles are congruent. We can draw a triangle with the apothem as one leg, the side length as the hypotenuse, and the bisector of an angle as the other leg. The bisector of an angle divides the angle into two congruent angles, so we can use the fact that the sum of the angles in a triangle is 180 degrees to find the measure of each angle. Each angle of the hexagon measures 120 degrees, so each angle of the triangle measures 60 degrees. Therefore, we can use the sine function to find the side length: sin 60 = 5β3 / s s = 5β3 / sin 60 s = 10β3 / 3 Now we can use the formula for the area of a regular hexagon: A = (3β3 / 2) * a^2 where a is the side length. Substituting s into the formula, we get: A = (3β3 / 2) * (10β3 / 3)^2 A = 150β3 Rounding to the nearest tenth, the area of the hexagon is approximately 259.8 square units.