How many rectangles can be drawn with perimeter of each rectangle as 16 cm?
the length+width must be 8.
So, how many pairs of numbers add to 8?
1+7
2+6
...
l = length
w = width
P = 2 ( w + l )
If each side is a whole number, then:
1.
w = 1 , l = 7
P = 2 ( w + l )
P = 2 ( 1 + 7 ) = 2 * 8 = 16
1 x 7
2.
w = 2 , l = 6
P = 2 ( w + l )
P = 2 ( 2 + 6 ) = 2 * 8 = 16
2 x 6
3.
w = 3 , l = 5
P = 2 ( w + l )
P = 2 ( 3 + 5 ) = 2 * 8 = 16
3 x 5
4.
w = 4 , l = 4
P = 2 ( w + l )
P = 2 ( 4 + 4 ) = 2 * 8 = 16
4 x 4
5.
w = 5 , l = 3
P = 2 ( w + l )
P = 2 ( 5 + 3 ) = 2 * 8 = 16
5 x 3
6.
w = 6 , l = 2
P = 2 ( w + l )
P = 2 ( 6 + 2 ) = 2 * 8 = 16
6 x 2
7.
w = 7 , l = 1
P = 2 ( w + l )
P = 2 ( 7 + 1 ) = 2 * 8 = 16
7 x 1
7 x 1 is same 1 x 7
6 x 2 is same 2 x 6
4 x 4 is square
5 x 3 is same 3 x 5
So 3 rectangles:
1 x 7
2 x 6
3 x 5
If each side isn't a whole number, than infinity rectangles can be drawn.
To find the number of rectangles that can be drawn with a perimeter of 16 cm, we can explore the different dimensions that satisfy this condition.
Let's assume the length of the rectangle is denoted by "l" and the width by "w".
Given that the perimeter of a rectangle is calculated by adding the lengths of all its sides, we can express the perimeter as:
Perimeter = 2l + 2w
Since the perimeter of our rectangle is 16 cm, we can write the equation as:
2l + 2w = 16
Simplifying the equation further, we divide both sides by 2:
l + w = 8
Now, we can explore the different combinations of lengths and widths that satisfy this equation.
For example:
- If the length, l, is 1 cm, the width, w, would be 7 cm.
- If l = 2 cm, w = 6 cm.
- If l = 3 cm, w = 5 cm.
We can continue trying different combinations until we find all possible pairs of lengths and widths. However, since rotations are allowed for rectangles, we must consider that switching the length and width of a rectangle with the same dimensions will result in the same rectangle.
By listing out all the possible combinations, we find the following pairs of lengths and widths:
{1, 7}, {2, 6}, {3, 5}, {4, 4}, {5, 3}, {6, 2}, {7, 1}
But there is an exception for the pair {4, 4} because it represents a square, not a rectangle with different length and width. Hence, counting the pairs without the square, we have a total of 6 rectangles.
Therefore, there are 6 rectangles that can be drawn with a perimeter of each rectangle as 16 cm.