How many squares of all sizes are there in a 5 by 5 grid of small squares?

Question

The largest sqare is the one on the outside. The smallest are the 25 making up the grid. I need the total number using all the sizes of squares in the box. Thank you!

Answer 1

You need a way to count without worrying about repetition.
First count all single squares, by counting the centres of the squares. You already said its 5x5.
Next count all 2x2 sqaures. Consider the possible centres of these squares. They can only lie on interior line crossings, of which there are 4x4=16.
For 3x3 squares, you get a 3x3 grid of possible centres, and so on. Cleary there is only one 5x5.

25+16+9+4+1.

Answer 2

The answer would be 1+2^2+3^2+4^2+5^2
=1+4+9+16+25
=55

Answer 3

I assume you just want squares (no rectangles).

The answer is:
25 squares of 1x1
16 squares of 2x2
9 squares of 3x3
4 squares of 4x4
1 square of 5x5

That's a total of 55 squares!

Answer 4

There are:
1 square 5x5
4 squares 4x4
9 squares 3x3
...