what is the area of a polygon with a apothem of 6

Answer 1

To answer this question, we need to know how many sides the polygon has and if it is a regular polygon. There is an easy formula for calculating the area, but it does involve knowing specifically what type of polygon it is. Were you given this information?

Ok, thanks for the info from your response to my question above. Now that we know the number of sides, we can get to work! (-: The formula for the area of a regular polygon is A = 1/2 a p, where a = the apothem and p = the perimeter. With 6 sides, we need to find the length of one side and then multiply by 6 to determine the perimeter. We will do this by setting up a diagram of the regular hexagon, then drawing in the apothem, then also drawing in a segment from the center of the polygon to a vertex of the polygon such that the apothem, the segment from the center to a vertex, and half of one of the sides of the polygon make a right triangle. The triangle formed will be a 30-60-90 right triangle with the 30 degree angle having its vertex at the center of the polygon and the right angle formed by the apothem and the half-side of the polygon.

Once the diagram is set up, we must now note that the apothem is also the longer leg of the right triangle, and in a 30-60-90 triangle, the longer leg = the shorter leg multiplied by the square root of 3. So to find the shorter leg of our triangle, we must divide: 6/radical 3. This simplifies to 2(radical 3). Note, however, that this measure is only half of one of the sides of the polygon, so a full side of the polygon would be twice this, or 4(radical 3). Since there are 6 sides, this would make the perimeter = 24(radical 3).

Now we can insert all the needed values into our area formula to finish the problem:

A = 1/2 (6)( 24 radical 3) which simplifies to 72 (radical 3), or 72 times the square root of 3.