Complete the hexagonal and star shaped Rangolies [see Fig. $ 7.53 $ (i) and (ii)] by filling them with as many equilateral triangles of side $ 1 \mathrm{~cm} $ as you can. Count the number of triangles in each case. Which has more triangles?
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Given:

hexagonal and star shaped rangolies.

To do:

We have to fill the given rangolies with as many equilateral triangles of side $1\ cm$ as we can and count the number of triangles in each case, which has more triangles.

Solution:

Let us calculate the area of hexagon and star,

Area of hexagon$= 6\times\frac{25\sqrt 3}{4}$

Area of star shaped rangoli$=12\times\frac{\sqrt 3}{4}\times5^2$

The area of equilateral triangles with $1\ cm$ side$=\frac{\sqrt 3}{4}\times a^2$

This implies,

$\frac{\sqrt 3}{4}\ cm^2$

Therefore,

The no.of equilateral triangles with $1\ cm$ will be $=\frac{6\times\frac{25\sqrt 3}{4}}{\frac{\sqrt 3}{4}\ cm^2}$

This implies,

The no of equilateral triangles in hexagon with $1\ cm$ side$=150$

Similarly,

The area of equilateral triangles with $1\ cm$ side $=\frac{12\times\frac{\sqrt 3}{4}\times5^2}{\frac{\sqrt 3}{4}\ cm^2}$

This implies,

The no of equilateral triangles in star shaped rangoli with $1\ cm$ side$=300$

Therefore,

The triangles with side $1\ cm$ in star shaped rangoli is more in number than in hexagonal shaped rangoli.

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Updated on: 10-Oct-2022

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