Answer:
maths stuff
Explanation:
To find the area of a regular hexagon with an apothem length of 4 centimeters, we can use the formula:
Area = (1/2) * apothem * perimeter
where "apothem" is the distance from the center of the hexagon to the midpoint of one of its sides, and "perimeter" is the total length of the hexagon's sides.
Since we know the apothem length, we need to find the length of one of the sides of the hexagon. To do this, we can use trigonometry.
Divide the hexagon into six congruent equilateral triangles, and draw a line from the center of the hexagon to the midpoint of one of the sides of the triangle, creating a right triangle. The hypotenuse of this right triangle is the length of one side of the hexagon, and the apothem is one of the legs. The angle between the apothem and the hypotenuse is 30 degrees, since it is half of the angle at the center of one of the triangles, which is 60 degrees.
Using the trigonometric function "tangent", we can find the length of the side:
tan(30 degrees) = side length / apothem
side length = apothem * tan(30 degrees)
side length = 4 cm * tan(30 degrees)
side length = 4 cm * 1/sqrt(3)
Now we can find the perimeter of the hexagon:
perimeter = 6 * side length
perimeter = 6 * 4 cm * 1/sqrt(3)
perimeter = 24/sqrt(3) cm
Finally, we can use the formula to find the area:
Area = (1/2) * apothem * perimeter
Area = (1/2) * 4 cm * 24/sqrt(3) cm
Area = 48/sqrt(3) cm^2
Therefore, the area of the regular hexagon with an apothem length of 4 centimeters is exactly 48/sqrt(3) square centimeters.