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Find the area of a regular hexagon with an apothem of 12 centimeters.

Sagot :

reedemily20011

Answer:

To find the area of a regular hexagon with an apothem of 12 centimeters, we can use the formula:

Area = 1/2 * Perimeter * Apothem

In a regular hexagon, each interior angle measures 120 degrees, and the apothem is the distance from the center of the hexagon to the midpoint of a side, forming a right angle with that side.

Given that the apothem is 12 centimeters, and in a regular hexagon, the apothem is also the height of each equilateral triangle formed within the hexagon, we can find the side length of the hexagon using trigonometry:

The apothem divides the equilateral triangle into two right-angled triangles. The height (apothem) is 12 cm, and the base (side of the hexagon) is the hypotenuse of one of these right-angled triangles.

Using trigonometry (sin 60° = opposite/hypotenuse), we can find the side length:

sin 60° = 12 / side length

side length = 12 / sin 60°

side length = 12 / √3 / 2

side length = 12 * 2 / √3

side length = 24 / √3

side length = 24√3 / 3

side length = 8√3

Now, we can calculate the perimeter of the hexagon:

Perimeter = 6 * side length

Perimeter = 6 * 8√3

Perimeter = 48√3

Finally, we can find the area using the formula:

Area = 1/2 * Perimeter * Apothem

Area = 1/2 * 48√3 * 12

Area = 24 * 12√3

Area = 288√3 square centimeters

Therefore, the area of the regular hexagon with an apothem of 12 centimeters is 288√3 square centimeters.

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