SOLUTION: A star shape is made from 6 congruent equilateral triangles and a regular hexagon. The star shape has an area of 96cm^2. What is the area of the regular hexagon?
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Question 1083475: A star shape is made from 6 congruent equilateral triangles and a regular hexagon. The star shape has an area of 96cm^2. What is the area of the regular hexagon? Found 2 solutions by jim_thompson5910, MathTherapy:Answer by jim_thompson5910(35256) (Show Source):
A drawing often helps solve the problem very easily
The red triangular exterior pieces are all equilateral triangles with some side length.
We don't need to know what that side length is. It doesn't matter.
The area of one red equilateral triangle is some area A.
There are 6 of these red triangles, so the red exterior triangular parts combine to a total area of 6*A.
The hexagon is composed of equilateral triangles as well.
Each of these blue equilateral triangles is congruent to any outer red triangle because the side length is the same.
Therefore, the 6 blue equilateral triangles composing this hexagon combine to get a total area of 6*A
The area of the star overall is 12*A because we have 6 blue triangles combining with the 6 red triangles.
There is a total of 12 triangles.
To recap so far:
Area of hexagon = 6*A
Area of star = 12*A
The ratio of the two areas is
R = (area of hexagon)/(area of star)
R = (6A)/(12A)
R = 1/2
R = 0.5
Basically telling us the area of the hexagon is half that of the star.
We can see this with a bit of algebra
R = (area of hexagon)/(area of star)
R*(area of star) = area of hexagon
area of hexagon = R*(area of star)
area of hexagon = 0.5*(area of star)
Now plug in the star's area
area of hexagon = 0.5*(area of star)
area of hexagon = 0.5*(96)
area of hexagon = 48 square centimeters
You can put this solution on YOUR website! A star shape is made from 6 congruent equilateral triangles and a regular hexagon. The star shape has an area of 96cm^2. What is the area of the regular hexagon?
Number of equilateral triangles contained in the star: 12
Area of each equilateral triangle:
Number of equilateral triangles contained in HEXAGON (6-sided polygon): 6
Area of the 6 equilateral triangles making up the hexagon:
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