What is the area of a regular hexagon with a distance from its center to a vertex of 1 cm? (Hint: A regular hexagon can be divided into six equilateral triangles.)
hex
The area of the regular hexagon
is _____ ?
well, we need the area of an equilateral triangle with sides of one cm. Then we multiply by six
for altitude
sin 30 = (1/2)/1
cos 30 = h/1
h=.866
so
area of triangle = (1/2)(1)(.866)
= .433
then multiply by six
6*.433 = 2.6 cm^2
To find the area of a regular hexagon, we can divide it into six equilateral triangles.
Each equilateral triangle has a side length equal to the distance from the center to a vertex of the hexagon, which in this case is 1 cm.
The formula for the area of an equilateral triangle is:
Area = (√3/4) * s^2
where s is the length of a side.
Since the side length of each equilateral triangle is 1 cm, we can substitute s = 1 into the formula:
Area = (√3/4) * (1)^2
Area = (√3/4) * 1
Area = (√3/4) cm^2
So the area of each equilateral triangle is (√3/4) cm^2.
Since there are six equilateral triangles in a regular hexagon, to find the total area of the hexagon, we multiply the area of one triangle by six:
Total area = 6 * (√3/4) cm^2
Total area = 6√3/4 cm^2
Thus, the area of the regular hexagon is 6√3/4 cm^2.
To find the area of a regular hexagon, we can use the fact that it can be divided into six equilateral triangles.
Step 1: Find the side length of the equilateral triangle.
Since the distance from the center to a vertex is given as 1 cm, this distance is also the height of the equilateral triangle. In an equilateral triangle, the height bisects the base, making two right-angled triangles with equal sides. Therefore, the base of each right-angled triangle is half the side length of the equilateral triangle.
Using Pythagoras' theorem, we can find the side length of the equilateral triangle:
(side length)^2 = (height)^2 + (base)^2
(side length)^2 = 1^2 + (0.5)^2
(side length)^2 = 1 + 0.25
(side length)^2 = 1.25
Taking the square root of both sides, we get:
side length ≈ √1.25 ≈ 1.12
Step 2: Find the area of one equilateral triangle.
The area of an equilateral triangle can be found using the formula:
Area = (side length^2 * √3) / 4
Plugging in the value of the side length we found earlier:
Area of one equilateral triangle = (1.12^2 * √3) / 4 ≈ 0.97 square cm
Step 3: Find the area of the regular hexagon.
Since the regular hexagon can be divided into six equilateral triangles, we can find its area by multiplying the area of one equilateral triangle by 6:
Area of the regular hexagon = 0.97 * 6 ≈ 5.82 square cm
Therefore, the area of the regular hexagon is approximately 5.82 square cm.