Answer:
Explanation:
To solve this problem, we can use the fact that a regular hexagon can be divided into six equilateral triangles.The radius of the hexagon is given as 12. Let's draw a diagram to better visualize the hexagon / \ / \ / \ /_______\ B C |\ /| | \ / | | \ / | | \ / | |____D____| | / \ | | / \ | | / \ | |/_______\| E FIn the diagram above, A, B, C, D, E, and F are the vertices of the hexagon.To find the apothem of the hexagon, we need to find the height of one of the equilateral triangles. We know that the side length of each triangle is equal to the radius of the hexagon, which is 12. Using the Pythagorean theorem, we can find the height of one of the triangles/| / | h /__| b/2 (b/2)^2 + h^2 = 12^2 b^2/4 + h^2 = 144 h^2 = 144 - b^2/4Since the hexagon is regular, each side length is equal to 12. Therefore, b = 12.scssCopy codeh^2 = 144 - 12^2/4 h^2 = 144 - 36 h^2 = 108 h = sqrt(108) h = 6*sqrt(3)So the apothem of the hexagon is 6*sqrt(3).To find the area of the hexagon, we can use the formula:makefileCopy codeArea = (1/2) * apothem * perimeterThe perimeter of the hexagon is equal to six times the length of one side, which is 6*12 = 72.scssCopy codeArea = (1/2) * 6*sqrt(3) * 72 Area = 216*sqrt(3)So the area of the hexagon is 216*sqrt(3).Finally, the length of one side of the hexagon is simply equal to the radius, which is 12.