Is Bob's method for determining the area of the irregular shape valid or not?

Question

Bob wants to find the area of an irregular shape. He cuts a piece of string to the length of the perimeter of the shape. He measures to see that the string is about 60 cm long. Bob then forms his string into a square on top of the centimeter graph paper. Using the graph paper, he determines that the area of his string square is about 225 cm^2. Bob says that the area of the irregular shape os therefore also 225 cm^2. Is Bob's method for determining the area of the irregular shape valid or not? IF the method is not valid, what can you determine about the area of the irregular shape from the information that Bob has?

Answer 1

Bob's method is incorrect. Here is a counterexample.

Measure a rectangle. The perimeter is 24 units.

Therefore the length plus the width is 12

Calculating the possible areas for some integer values of the length and width:

1 and 11, area = 11

2 and 10, area = 20

3 and 9, area = 27

4 and 8, area = 32

and so on.

 

The area of an irregular shape cannot be determined from solely the length of the perimeter.