We all know area formula for various standards shapes, but have you ever wondered how these would have been derived or at least tried to verify it? I came across this thought while observing beautiful design works on the floor of a temple and tried my hands to find the area of somewhat bigger hexagon than the one shown below. You too try it out and post your finding in the comments section. Let us try to learn the basics, as basics are vital to start revolutions!!!!!!
Consider each
to be one 1 sq meter for our convenience and prove the area formulae for above hexagon.So our hexagon has 3
on a side, and it is a regular hexagon. Regular hexagon is made up of 6 equilateral triangle and area formula for an equilateral triangle is (sqrt (3)/4)*(side) ^2. Hence our formula answer is 6*((sqrt (3)/4)*(side) ^2) = 23.38, but there are only 19 smaller hexagon. Could you identify where I have gone wrong in my calculation?Example: Here you can see how formulae for calculating area of a square are verified.
In subsequent posts, let’s divert our attention on how these formulas were derived. Happy learning!!!
i don't get your problem fully. Are you trying to say that the area of the larger hexagon should be = 19* area of smaller hexagon = 6 * 19 * area of smaller equilateral triangles?
ReplyDeleteWell, for starters, area of the larger hexagon should be != 19* area of smaller hexagon because of the gaps between the hexagons.
Please explain more precisely.
@ palindromic existence
ReplyDeleteMy motive was to find out relation between the number of smaller fixed shape( here smaller hexagon) and bigger regular shape ( here bigger hexagon and square) areas.
This question arose as part of my trial to find out how do people say that a rectangle is of so much sq units?
In one of my math GRE class, my tutor who is a prof at New college, explained that so many squares of area 1 could be fit inside it.
Here i tried to find out number of smaller shapes within a regular bigger shape using that principle by assuming the smaller fixed shape to be the standard unit.
Please find my friend aditya's explanation below
ReplyDeleteAccording to the blog you want the area of the red hexagon i.e. entire figure.
This area is the sum of the hexagons + sum of the inner triangles.
There are 19 hexagons and 33 identical triangles (check them!)
If the hexagon has unit area, then each triangle has area 1/6.
This is because the triangles are all equilateral and identical to the six equilateral triangles that cover a unit hexagon (these triangles can be constructed by joining the corners of the hexagon to the centre).
So the answer is 19 + 33/6
For the square grid of hexagons, you seem to have confused unit area hexagons with side length of the square. If you analyse the area problem carefully like for the hexagon above, you can get the area of the square grid correctly. (It is more than 9 sq. units.).
@ aditya
ReplyDeleteexactly adi, i figured this one out --> the smaller inner triangles.
But i got confused when analysing the same pattern for square shape which i had given below as an example.
Could you share your thoughts about where i had gone wrong with square example?