What is the perimeter of a regular hexagon that has an area of #54sqrt3# units squared?
3 Answers
The perimeter of the regular hexagon is
Explanation:
The formula for the area of a regular hexagon is
regular hexagon.
The perimeter of the regular hexagon is
unit. [Ans]
Perimeter:
Explanation:
A hexagon can be decomposed into 6 equilateral triangles:
If we let
The area of a triangle with sides of length
The area of the hexagon is
The perimeter of the hexagon is
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Finding the perimeter of an equilateral triangle with sides of length
Heron'[s formula for the area of a triangle tells us that if the semi-perimeter of a triangle is
The semi-perimeter is
So
and
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Explanation:
Let's start from an equilateral triangle with side
Bisecting the triangle results in two right angled triangles, with sides
#1^2 + (sqrt(3))^2 = 2^2#
The area of the equilateral triangle is the same as a rectangle with sides
Six such triangles can be assembled to form a regular hexagon with side
In our example, the hexagon has area:
#54 sqrt(3) = color(blue)(3)^2 * (6 sqrt(3))#
So the length of each side is:
#color(blue)(3) * 2 = 6#
and the perimeter is:
#6 * 6 = 36#