How to find the area of a regular hexagon with a perimeter of 78 inches and a perpendicular distance of 11 inches from side to center

To find the area of a regular hexagon, you can use the formula:

Area = (3√3 / 2) * s^2

Where "s" is the length of each side of the hexagon.

In this case, you are given the perimeter of the hexagon (78 inches). Since a regular hexagon has six equal sides, you can determine the length of each side by dividing the perimeter by 6:

s = perimeter / 6
s = 78 inches / 6
s = 13 inches

Now, you need to find the perpendicular distance from one side to the center of the hexagon (11 inches). This distance is known as the apothem.

To calculate the area, substitute the values of "s" and "apothem" into the formula:

Area = (3√3 / 2) * (s^2)
Area = (3√3 / 2) * (13 inches)^2
Area = (3 * 1.732 / 2) * 169 inches^2
Area = 259.8079 inches^2

Therefore, the area of the regular hexagon is approximately 259.8079 square inches.

look at one of the 6 congruent triangles

the base of it must be 78/6 = 13
area of one of them = (1/2)base x height
= (1/2)(13)(11) = 143/2 square inches

so the whole hexagon is 6(143/2) = 429 inches^2