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Complete the hexagonal and star shaped Rangolies [see Fig. (i) and (ii)] 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?

4.  Complete the hexagonal and star shaped Rangolies [see Fig. (i) and (ii)] by filling them with as many equilateral triangles of side \small 1 cm as you can. Count the number of triangles in each case. Which has more triangles?

                                

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For finding the number of triangles we need to find the area of the figure.

Consider the hexagonal structure :

                      Area of hexagon  =   6  \times   Area of 1 equilateral 

Thus area of the equilateral triangle :           

           =\ \frac{\sqrt{3}}{4}\times a^2

or        =\ \frac{\sqrt{3}}{4}\times 5^2

or        =\ \frac{25\sqrt{3}}{4}\ cm^2

So, the area of the hexagon is  : 

       =\6\times \frac{25\sqrt{3}}{4}\ =\ \frac{75\sqrt{3}}{2}\ cm^2

And the area of an equilateral triangle having 1cm as its side is :

       =\ \frac{\sqrt{3}}{4}\times 1^2

or    =\ \frac{\sqrt{3}}{4}\ cm^2

Hence a number of equilateral triangles that can be filled in hexagon are :     

    =\ \frac{\frac{75\sqrt{3}}{2}}{\frac{\sqrt{3}}{4}}\ =\ 150

Similarly for star-shaped rangoli :

 Area :             

   =\12\times \frac{\sqrt{3}}{4}\times 5^2 \ =\ 75\sqrt{3}\ cm^2

Thus the number of equilateral triangles are : 

      =\ \frac{75\sqrt{3}}{\frac{\sqrt{3}}{4}}\ =\ 300

Hence star-shaped rangoli has more equilateral triangles.

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