What is the area of a rectangle with a perimeter of 16cm?

Any area you want greater than 0 cm2 and less than or equal to 16 cm2.

perimeter_of_rectangle = 2 x (width + length)

16 cm = 2 x (width + length)

⇒ width + length = 8 cm

⇒ length = 8 cm - width

Neither dimension can be negative, so they are both greater than 0 cm and less than 8 cm.

Area_of_rectangle = width x length

⇒ area = width x (8 cm - width)

= 8 x width - width2 cm2

This has a minimum value when width = 0 cm or 8cm of 0 cm2, so the area of the perimeter must be greater than 0 cm2 (as the width can be neither 0 cm nor 8 cm).

This has a maximum value when width = 4 cm:

max_area = 8 x 4 - 42 cm2 = 16 cm2.

when the width is between 0 cm and 4 cm (and so the length is between 8 cm and 4 cm) the area is between 0 cm2 and 16 cm2

When the width = 4 cm, the length is also 4 cm, that is the rectangle is a square.