find the apothem, in meters, and area, in square meters, of a regular hexagon with side length 7 m. round to the nearest hundredth
the hexagon is composed of six equilateral triangles
the apothem is the altitude of one of the triangles
the area is the sum of the triangles' areas
Concentrate on just one of the six congruent isosceles triangles
For each one:
the central angle is 60° and the two base angles are equal,
ahhh, so each triangle is equilateral.
each side in such a triangle is 7 m
The apothem is the perpendicular to one of the sides, let's call it a
a/7 = sin60
a = 7(√3/2) = ....
area of 1 triangle = (1/2)(base)(height)
= (1/2)(7)(7√3/2) = ....
so the whole hexagon is 6 times that area
To find the apothem (a) of a regular hexagon, you can use the formula:
a = (s / 2) / tan(π/6)
where s represents the side length of the hexagon.
Given that the side length (s) is 7 m:
a = (7 / 2) / tan(π/6)
Using the value of π as 3.14 and evaluating tan(π/6), we can calculate the apothem:
a = (7 / 2) / tan(π/6)
≈ (7 / 2) / 0.577
≈ 6.07
Therefore, the apothem of the hexagon is approximately 6.07 meters.
To find the area (A) of the regular hexagon, you can use the formula:
A = (3√3 / 2) * s^2
where s represents the side length.
Given that the side length (s) is 7 m:
A = (3√3 / 2) * 7^2
Using the value of √3 as approximately 1.732, we can calculate the area:
A = (3√3 / 2) * 49
≈ (3 * 1.732 / 2) * 49
≈ 121.144
Therefore, the area of the hexagon is approximately 121.144 square meters, rounded to the nearest hundredth.
To find the apothem and area of a regular hexagon, we can use the following formulas:
1. Apothem (a): The apothem is the distance from the center of the hexagon to the midpoint of any side.
a = (s/2) * tan(π/n)
Where s is the side length of the hexagon, and n is the number of sides (in this case, 6 for a hexagon).
2. Area (A): The area of a regular hexagon can be found by multiplying the apothem by the perimeter and dividing by 2.
A = (P * a) / 2
Where P is the perimeter of the hexagon, and a is the apothem.
Now let's calculate the apothem and area of the hexagon with side length 7 m:
1. Apothem (a):
a = (7/2) * tan(π/6)
a = (7/2) * tan(π/6)
a = (7/2) * tan(π/6)
a = (7/2) * 0.57735
a ≈ 6.123 m (rounded to the nearest hundredth)
2. Area (A):
The perimeter of a regular hexagon is simply 6 times the side length.
P = 6 * 7
P = 42 m
A = (42 * 6.123) / 2
A = 257.646 / 2
A ≈ 128.823 m² (rounded to the nearest hundredth)
Therefore, the apothem of the hexagon is approximately 6.12 m, and the area of the hexagon is approximately 128.82 m² (rounded to the nearest hundredth).