Area of a Regular Polygon?

Question

I am not certain of my answers, and I would greatly appreciate if you could check them for me. Any help is great!

 

The question is what is the area of the regular polygon with the given radius or apothem? If your
answer is not an integer, leave it in simplest radical form.

 

Question 1) The first shape is a square with an apothem of 6. I figured that since the sides are equal each side equaled double the apothem and amounted to 12. Then I did 12*4 = 48. 48 * 48 = 2304, which made area = 2304 cm^2. 

 

Question 2) The second shape is a hexagon with an apothem of 8* the square root of 3. To solve this I did 360 degrees divided by the 6 sides. Then tan(30 degrees) = x / 8 (square root of 3).

 

X = 8
8*2 = 16
96 * 8(square root of 3) = 1330.2
Area = 665.12 in. 

 

Question 3) The third shape is another hexagon with an apothem of 6* the square root of three. To solve this I did 360 degrees divided by the 6 sides. Then tan(30 degrees) = x / 6 (square root of 3).
X = 6
6*2 = 12
72 * 6(square root of 3) = 748.3
Area = 374.1m

Answer 1

Happy to check these. Two of them are right and one has a slip in it, so let us go through them.

The formula that handles every regular polygon at once is:

A = (1/2)(apothem)(perimeter)

It is worth seeing where that comes from. Slice the polygon from the center into triangles, one per side. Each triangle has the side as its base and the apothem as its height, so each has area (1/2)(side)(apothem). Adding them all up, the sides combine into the perimeter and you get the formula above.

Question 1, the square with apothem 6. Your first step is right. In a square the apothem reaches from the center to the middle of a side, so the full side is 2 times 6 = 12, and the perimeter is 48. But then the area got mixed up. You squared the PERIMETER instead of the side:

A = (1/2)(6)(48) = 144

or just as easily, side times side, 12 x 12 = 144. So it is 144 cm^2, not 2304.

Question 2, the hexagon with apothem 8sqrt(3). Your setup with the 30 degree angle is exactly right, and you correctly got a half side of 8, so the full side is 16 and the perimeter is 96:

A = (1/2)(8sqrt(3))(96) = 384sqrt(3)

Remember, the problem asked for simplest radical form when the answer is not an integer. So the answer to write down is 384sqrt(3) in^2 rather than the decimal. Your 665.12 is that same number rounded, so your arithmetic was fine, it just needs to stay exact.

Question 3, the hexagon with apothem 6sqrt(3). Same method, side of 12 and perimeter of 72:

A = (1/2)(6sqrt(3))(72) = 216sqrt(3)

Again, leave it as 216sqrt(3) m^2 instead of 374.1.

So, our final answers are 144 cm^2, 384sqrt(3) in^2, and 216sqrt(3) m^2. Your method was solid on all three, the only real error was squaring the perimeter on the square, and the rest is just remembering to leave radicals exact when the problem asks for it.

Answer 2

in the case of questions 2 and 3, two possibilities exist, namely regular or irregular hexagons. 

a regular hexagon is composed of 6 equilateral triangles, in which case, the apothem length is the height of each triangle as it is perpendicular to a side of the hexagon.
As such, it forms a 30-60-90 right triangle.

We can then calculate the length of a triangle side by
Sin 60o = apothem length/hypotenuse
Sin 60o = (8√3)/h
0.8 = (8√3)/h
h = (8√3)/0.8     h = 17.3

with the height of 8√3, and a side of the triangle at 17.3, we can then calculate the area of the hexagon using the area of a triangle multiplied by 6
in this case side = 17.3, and height = 8√3, which is 13.85
thus, the area of each triangle will be ½(17.3 * 13.85) = 120
so, the area of the hexagon will be 6 *120 = 720 units

in like manner, for problem 3, the apothem is 6√3 (10.4)
sin 60o = 10.4 / side
0.80 = 10.4 / s
s = 10.4/0.8 = 13
area = 6 [1/2(13 * 10.4)] = 6 * 67.6 = 406 units


if, however, the hexagons are irregular, then more information will be necessary to solve the problems. that is, enough information to define the 6 triangles which form the hexagon

Answer 3

Rachel,

         Not sure what you are doing on # 2 and 3. On question 1 however, the apothem =6 so the side is 2*6=12 as you stated but the area of a square is equal to one side squared  A=s² so the area is 12² or 144 sq units.

 

Hope this helps

 

Jim