Question

Question 4
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?

Answer 1

We may observe that hexagonal shaped rangoli is having 6 equilateral triangles in it.
Area of $\Delta$ OAB = $\frac{\sqrt{3}}{4}(side{)}^2=\frac{\sqrt{3}}{4}(5{)}^2$
Area of hexagonal shaped Rangoli = $6\times \frac{25\sqrt{3}}{4}=\frac{75\sqrt{3}}{4}c{m}^2$
Area of Equilateral Triangle of 1cm = $\frac{\sqrt{3}}{4}c{m}^2$
Number of Equilateral triangles of 1cm side that can be filled in hexagonal shaped rangoli = 150
Star shaped Rangoli is having 12 equilateral triangles of side 5cm, in it

Area of star-shaped rangoli = $12\times \frac{\sqrt{3}}{4}(5\times 5)=75\sqrt{3}$
Number of equilateral triangles of 1 cm side that can be filled in this shaped rangoli = 300
So, star-shaped rangoli has more equilateral triangles in it.