Ex 7.5, 4 (Optional)
Complete the hexagonal and star shaped Rangolies [See Fig 7.53 (i) and (ii) by filing 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?
Let’s look at the Hexagon first
Hexagon
Let’s join the vertices of the hexagon to its center
By symmetry, we can see that
there are 6 equilateral triangles with side 5 cm
Let’s find how many equilateral triangle of side 1 cm can be fit inside the equilateral triangle of side 5 cm
Finding equilateral triangles in 5 cm equilateral triangle
Making 1 cm equilateral triangles in the triangle
We see that
Total number of 1 cm equilateral triangles
= 1 + 3 + 5 + 7 + 9
= 25
We can also find number of triangles by area
Area of 1 cm equilateral triangle = √3/4 (Side)2
Area of 1 cm equilateral triangle = √3/4 × 12 = √3/4 × 1
Area of 5 cm equilateral triangle = √3/4 × 52 = √3/4 × 25
Therefore,
Number of 1 cm equilateral triangles
= (𝐴𝑟𝑒𝑎 𝑜𝑓 5 𝑐𝑚 𝑒𝑞𝑢𝑖𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒)/(𝐴𝑟𝑒𝑎 𝑜𝑓 1 𝑐𝑚 𝑒𝑞𝑢𝑖𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒)
= (√3/4 \ × 25)/(√3/4 \ × 1) = 25
Now,
Hexagon has 6 equilateral triangle of side 5 cm
So,
Number of equilateral triangles of side 1 cm in hexagon
= 6 × 25
= 150
Rangoli
Rangoli has a hexagon in the middle
And 6 equilateral triangles of 5 cm
Thus,
Total number of equilateral triangles of side 5 cm
= 6 + 6
= 12
So,
Number of equilateral triangles of side 1 cm in Rangoli
= 12 × 25
= 300
So, Hexagon as 150 triangles
Rangoli has 300 triangles

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

Hi, it looks like you're using AdBlock :(

Displaying ads are our only source of revenue. To help Teachoo create more content, and view the ad-free version of Teachooo... please purchase Teachoo Black subscription.

Please login to view more pages. It's free :)

Teachoo gives you a better experience when you're logged in. Please login :)

Solve all your doubts with Teachoo Black!

Teachoo answers all your questions if you are a Black user!