An equilateral triangle and a regular hexagon have equal perimeters. if the area of the triangle is 2, what is the area of the hexagon?

Question

Jayden Jackson · Accepted Answer

#A_h=3 " units"^2#

!
#=> A_t=sqrt3/4*(2x)^2=sqrt3/4*4x^2=2#
#=> x^2=2/sqrt3 " units"^2#

A regular hexagon can be divided into 6 congruent equilateral triangles, as shown in the figure.
given that the equilateral triangle and the regular hexagon have equal perimeter,
#=># side length of the hexagon #= (3*2x)/6=x# units
#=># area of the regular hexagon #=A_h=6*sqrt3/4*x^2#
#=> A_h=6*sqrt3/4*2/sqrt3=3 " units"^2#

Sadie Adams · Answer

undefined To find the area of the regular hexagon, we need to know the side length of both the equilateral triangle and the regular hexagon. Since they have equal perimeters, we can set up an equation using their respective formulas for perimeter.

For an equilateral triangle, the perimeter (P) is given by P = 3s, where s is the length of each side.

For a regular hexagon, the perimeter (P) is given by P = 6s, where s is the length of each side.

Since the perimeters are equal, we can equate the expressions for the perimeters:

3s = 6s

Now, we solve for s:

3s = 6s
3s - 6s = 0
-3s = 0
s = 0

This implies that the side length (s) of both shapes is zero, which is not feasible. Therefore, there seems to be an error in the given information. If we assume that the perimeters are not equal, we cannot directly determine the area of the regular hexagon based solely on the area of the equilateral triangle. Additional information about the relationship between the side lengths of the shapes or their perimeters would be required to solve the problem accurately.