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Perimeter is a length, and a length cannot be the same as an area.
Ignoring the units, all rectangles that have the same numerical perimeter as area are those that satisfy:
2 x (length + width) = length x width
which can be rearranged to give:
width = 2 x length/(length - 2)
Meaning that given any length over 2, a width can be found to give a rectangle that meets the requirement that its numerical perimeter is the same as its numerical area. (For a length greater than 0 and less than 2, the width would be negative and not possible; similarly for a length less than 0, the length is negative and not possible. When the length is 2, the width is undefined and so not possible. When the length is 0 the width is 0 and it is not a rectangle.)
At some stage as the length increases, the length will equal the width and as the length continues to increase the rectangle then given will match the previous rectangles with the length and widths swapped. This occurs when:
length = 2 x length/(length - 2)
⇒ length x (length - 4) = 0
⇒ length = 4.
So, as long as the length is greater than or equal 4 it will be the longer side - the length, by convention, is the longer side. Thus all rectangles satisfy:
width = 2 x length/(length - 2)
with length ≥ 4, will have the numerical value of their perimeter the same as the numerical value of their area.
For example:
- 4 cm x 4 cm: perimeter = 16 cm, area = 16 cm2
- 5 cm x 31/3 cm: perimeter = 162/3 cm, area = 162/3 cm2
- 6 cm x 3 cm: perimeter = 18 cm, area = 18 cm2
- 41/2 cm x 33/5 cm: perimeter = 161/5 cm, area = 161/5 cm2
- etc
Note: the first example is a square which is a rectangle with all the sides the same length.
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If two rectangles have the same area do they also have to have the same perimeter?
Not necessarily. Let's say that there is a circle with the area of 10. Now there is a star with the area of 10. They do not have the same perimeter, do they? That still applies with rectangles. There might be a very long skinny rectangle and a square next to each other with the same area, but that does not mean that they have the same perimeter. Now if the rectangles are congruent then yes.
Can different rectangles have the same area and perimeter?
It's very easy for two rectangles to have the same area and different perimeters,or the same perimeter and different areas. In either case, it would be obvious toyou when you see them that there's something different about them, and theywould not fit one on top of the other.But if two rectangles have the same area and the same perimeter, then to look at themyou'd swear that they're the same rectangle, and one could be laid down on the otherand fit exactly.
Can you Find 2 rectangles whose dimensions are whole nos and their area and perimeter is same?
4 x 4 and 6 x 3
What is the total perimeter of 6 square feet?
You can't tell the perimeter from the area. There are an infinite number of different shapes,all with different perimeters, that have the same area. Even if you only consider rectangles,there are still an infinite number of those that all have the same area and different perimeters.Here are a few rectangles with area of 6 square feet:Dimensions ... Perimeter0.75 x 8 . . . . . . 17.51 x 6 . . . . . . . . 141.5 x 4 . . .. . . . 112 x 3 . . . . . . . . 10
What are length and width of a rectangles sides which has an area of eighteen meters squared and a perimeter of eighteen meters?
Area can never be as low as the perimeter value -- impossible question.No impossible if read correctly...18 meters squared is the area and 18 meters is the perimeter.
