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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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