the area of the square and a rectangle are equal. The perimeter of the rectangle is 26cm. What is its area?

Answer :

Given  :  Area of square  =  Area of rectangle .

Let side of square  = x  , So,

Area of square  = x²

Let length of rectangle = a cm and width = b cm

Area of rectangle = a $\times$ b

Perimeter of  rectangle =  26 cm

So,

2 ( a + b  ) =  26                  [ Perimeter of rectangle =  2 (length + breadth)]

so, a  +  b  =  13 

Here we know sum of both sides of rectangle  =  13  ,

Now we take only integers value as length of side , and form different set as,

If length =  12 cm and width  =  1 cm  ,
So, Area of rectangle  =  12 $\times$ 1  =  12  cm²



If length =  11 cm  and width  =  2 cm  ,
So, Area of rectangle  =  11 $\times$ 2  =  22  cm²


If length =  10 cm and width  =  3 cm ,
So, Area of rectangle  =  10 $\times$ 3  =  30  cm²


If length =  9 cm and width  =  4 cm  ,
So, Area of rectangle  =  9 $\times$ 4  =  36  cm²


If length =  8 cm and width  =  5 cm  ,
So, Area of rectangle  =  8 $\times$ 5  =  40  cm²


If length =  7 cm and width  =  6 cm  ,
So, Area of rectangle  =  7 $\times$ 6  =  42  cm²

After that we get combination that gives same area as above we show in different length and width  .

Area of square could be  found by giving different integers value to x, we get

1² = 1 cm²
2² = 4 cm²
3² = 9 cm²
4² = 16 cm²
5² = 25 cm² 
6² = 36 cm²
7² = 49 cm²

Now we can see that in both area , we can have only one common area , that is 36 .

So,

Area of rectangle  =  Area of square  =  36 cm²                                                   ( Ans )