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Question
Complete the hexagonal and star shaped rangolies (see the given figures) by filling them with as many equilateral triangles of side 1 cm as you can. Count the number of triangles in each case. Which has more triangles?
Solution
It can be observed that hexagonal-shaped rangoli has 6 equilateral triangles in it.
`"Area of "ΔOAB = sqrt3/4"(side)"^2`
`= sqrt3/4(5)^2`
`= sqrt3/4(25)`
`= (25sqrt3)/4cm^2`
`"Area of hexagonal-shaped rangoli "=6xx(25sqrt3)/4=(75sqrt3)/2cm^2`
`"Area of equilateral triangle having its side as 1cm "=sqrt3/4(1)^2=sqrt3/4cm^2`
`"Number of equilateral triangles of 1 cm side that can be filled in this hexagonal-shaped rangoli "=((75sqrt3)/2)/(sqrt3/4)=150`
Star-shaped rangoli has 12 equilateral triangles of side 5 cm in it.
`"Area of star-shaped rangoli "=12xxsqrt3/4xx(5)^2=75sqrt3`
`"Number of equilateral triangles of 1 cm side that can be filled in this star-shaped rangoli "=(75sqrt3)/(sqrt3/4)=300`
Therefore, star-shaped rangoli has more equilateral triangles in it.
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