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Q: How many rectangles do you see in a 3 by 6 grid?

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36

90

the answer is 4 + 4 + 4 or 3 + 3 + 3 + 3

2 triangles and 3 rectangles(:

20

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14

36

90

There are 36 rectangles of various sizes formed by a 9-rectangle grid. The most obvious are the smallest (9) and the largest (1), but any 2, 3, 4, or 6 adjacent rectangles can form other rectangles.

the answer is 4 + 4 + 4 or 3 + 3 + 3 + 3

16 1x1 rectangles + 12 2x1 rectangles + 8 3x1 rectangles + 4 4x1 rectangles + 12 1x2 rectangles + 9 2x2 rectangles + 6 3x2 rectangles + 3 4x2 rectangles + 8 1x3 rectangles + 6 2x3 rectangles + 4 3x3 rectangles + 2 4x3 rectangles + 4 1x4 rectangles + 3 2x4 rectangles + 2 3x4 rectangles + 1 4x4 rectangle. A Grand Total of: 100 squares and rectangles. OR: A rectangle is formed by 2 horizontal lines and 2 vertical lines. There are 5 horizontal and 5 vertical lines so the number of rectangles is 5C2 * 5C2 = 10 * 10 = 100

2 triangles and 3 rectangles(:

In a grid of A x B squares, the formula to find how many unique rectangles there are (and all squares are considered to be rectangles) is: A * (A+1) * B * ((B+1)/4) A and B are interchangeable. So in a 5 x 4 grid, there are 5 * (5+1) * 4 * ((4+1)/4) Or 5 * 6 * 4 * (5/4) Or 150 unique rectangles. Now if we switch A and B, the equation reads: 4 * (4+1) * 5 * ((5+1)/4) Or 4 * 5 * 5 * (6/4) Again 150 unique rectangles.

4 times 4 equal 16. There are 16 individual rectangles in such a grid. This can however be a trickquestion. You may also have to think about all the rectangles that can be made with more than one individual rectangle. OOOO OOOO OOOO OOOO This illustration show the rectangles as O. It is crude, but understandable. If combining rectangles 3 by 3, then you also have 4 rectangles within the big rectangle that also should count as one.. Now we have 16+4+1 We should also count all rectangles made 2 by 2. This makes yet another 9 rectangles. All together, this makes 30 different rectangles. I am a bit unsure if a rectangle is a perfect square or not. If the rectangles does not have to be perfect squares, then the possible choices are even more. About 94 would then be my guess, but I have not rechecked this estimation.

There are 9 boxes in a 3x3 grid.

6

how many vertices does 3 hexagons and 8 rectangles have all together

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