How many small triangles are in a triangular grid that has 20 units on a side. Please show me a formula to figure it out . Thanks
Draw a few small grids. You can see that you get
#units #triangles
2 1
3 3
4 5
5 7
...
each new row adds two triangles in its length
and as you may recall, the sum of the first n odd numbers is n^2
So, there are (n-1)^2 triangles in a grid n units on a side.
Don't get it
Ah, triangles within triangles! Always a puzzling sight, but fear not, dear inquirer! I shall guide you through this mathematical maze with a dash of humor.
Now, to calculate the number of small triangles in a triangular grid with 20 units on each side, we shall employ a delightful formula. Prepare yourself, brave explorer:
The number of small triangles within a triangular grid of side length 'n' can be determined by the formula:
(num_triangles) = (n^2 + n) * (3n - 1) / 2
Let's plug in the value of 'n' as 20 and see what concoction our formula conjures:
(num_triangles) = (20^2 + 20) * (3 * 20 - 1) / 2
(num_triangles) = (400 + 20) * (60 - 1) / 2
(num_triangles) = 420 * 59 / 2
(num_triangles) = 24,810
Voila! We have a total of 24,810 small triangles in this particular triangular grid. Now you can conquer this perplexing puzzle with humor and confidence!
To answer this question, we can use the formula for the number of small triangles in a triangular grid.
The formula is n(n+1)(n+2)/2, where n represents the number of units on each side of the triangular grid.
In this case, n = 20, since the triangular grid has 20 units on a side.
Therefore, substituting n = 20 into the formula, we can calculate the number of small triangles as follows:
Number of small triangles = 20(20+1)(20+2)/2
= 20 * 21 * 22 / 2
= 9240
Hence, there are 9240 small triangles in the triangular grid with 20 units on each side.