Question

r 12 . Area of Shapes Class Activity 12T How Are Perimeter and Area Related for All Shapes? CCSS SMP3, SMP5 String would be helpful for this activity. Nick wants to find the area of the irregular shape below. He cuts a piece of string so that it goes all the way around the outside of the shape and then forms his piece of string into a square on top of graph paper. Using the graph paper, Nick gets a good estimate for the area of his string square and then uses the square's area as his estimate for the area of the original irregular shape Discuss whether Nick's method is a legitimate way to estimate areas of irregular shapes 1.

Answer 1

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

Measure a rectangle. Suppose the perimeter is 24 units.

Therefore the length plus the width is 12 since two times( length plus width) is perimeter.

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.

So as you can see the perimeter of the figure remains the same but its area keeps on changing continuously. The same can be true with our case also as Nick has measured the perimeter of the irregular figure by using a string. (now perimeter has been fixed).Now he takes this string and places on the graph paper to calculate the area formed by this string.so again the different possibility arises where the perimeter is the same but the area can be different as shown above.