Jody wants to cover her regular hexagon shaped table with a protective tablecloth. In order to do buy a perfectly-fit tablecloth she must know the area of the table. Using the given measure below, determine the remaining information.
Side Length: 12 cm
Apothem: 6√3 cm
WHats the area? and whats the permiter?
+4622 Let's see if I can attack both of the questions.
(a): The area of a regular hexagon can be found by splitting the hexagon into six equilateral triangles, using 30-60-90 triangles by drawing an altitude and multiply this results by six to get the area answer. However, a very popular formula for a hexagon is \(\frac{3\sqrt{3}}{2}s^2\) where \(s\) is the side length of the hexagon, and we will get an area(answer) of \(\boxed{216\sqrt{3}}\) centimeters squared.
-We can also solve this first part of the problem by noting the area of a regular figure or shape is half its perimeter times the distance from the center of the polygon to a side. (Try to prove this on your own!) And, the apothem is the distance from the center of a polygon to a side. We can generate and generalize the formula \(\frac{1}{2}*P*A\), where \(P\) is the perimeter and \(A\) is the apothem. Next, the area of the hexagon is \(12*6=72cm\), because the side length of the hexagon is twelve centimeters and a regular hexagon has six equal sides. We have the apothem as \(6\sqrt{3}\) since given in the problem. Thus,l the area is \(\frac{1}{2}*72*6\sqrt{3}=\boxed{216\sqrt{3}}\) centimeters squared.
(b): The perimeter is quite easier than calculating the area since the side length that is given in the problem is twelve centimeters, and a regular hexagon has six equal sides. Thus, the perimeter is \(12*6=72\)centimeters.
-tertre
+37153 A hexagon has 6 sides each side is 12 cm so perimeter= 12 * 6 = 72 cm^2
Area of hexagon 3sqrt3 /2 * a^2 where a = side length
3 sqrt3 144 /2 = 216 sqrt 3 sq cm
+4622 Let's see if I can attack both of the questions.
(a): The area of a regular hexagon can be found by splitting the hexagon into six equilateral triangles, using 30-60-90 triangles by drawing an altitude and multiply this results by six to get the area answer. However, a very popular formula for a hexagon is \(\frac{3\sqrt{3}}{2}s^2\) where \(s\) is the side length of the hexagon, and we will get an area(answer) of \(\boxed{216\sqrt{3}}\) centimeters squared.
-We can also solve this first part of the problem by noting the area of a regular figure or shape is half its perimeter times the distance from the center of the polygon to a side. (Try to prove this on your own!) And, the apothem is the distance from the center of a polygon to a side. We can generate and generalize the formula \(\frac{1}{2}*P*A\), where \(P\) is the perimeter and \(A\) is the apothem. Next, the area of the hexagon is \(12*6=72cm\), because the side length of the hexagon is twelve centimeters and a regular hexagon has six equal sides. We have the apothem as \(6\sqrt{3}\) since given in the problem. Thus,l the area is \(\frac{1}{2}*72*6\sqrt{3}=\boxed{216\sqrt{3}}\) centimeters squared.
(b): The perimeter is quite easier than calculating the area since the side length that is given in the problem is twelve centimeters, and a regular hexagon has six equal sides. Thus, the perimeter is \(12*6=72\)centimeters.
-tertre
