A and B are two similar 2D shapes. The area of shape A, which measures 12 cm, is 200 cm². Calculate the area of shape B, which measures 15 cm.

To calculate the area of shape B using the information provided, we need to establish the relationship between the linear dimensions of the two similar shapes and how these dimensions affect their respective areas.

Since shapes A and B are similar, the ratio of their corresponding linear dimensions is constant. Let's call the dimension of A that measures 12 cm the "base" of shape A, and let's say that the corresponding base of shape B is the part that measures 15 cm.

Now, we can set up a ratio of their corresponding bases: base of A / base of B = 12 cm / 15 cm

This can be simplified to express the scale factor between the two shapes: scale factor = 12/15 = 4/5

When dealing with similar shapes, the scale factor for the area is the square of the scale factor for the corresponding linear dimensions (because area is a two-dimensional measurement). Therefore:

scale factor for area = (scale factor for length)^2 = (4/5)^2 = 16/25

Shape A has an area of 200 cm². To determine the area of shape B using this scale factor, we can inverse the area scale factor to reflect the enlarged size of shape B as follows:

area of B / area of A = (scale factor for area of B to A)^2 area of B / 200 cm² = (5/4)^2 area of B / 200 cm² = 25/16

Now, solve for the area of B:

area of B = 200 cm² * (25/16) area of B = (200 * 25) / 16 area of B = 5000 / 16 area of B = 312.5 cm²

Therefore, the area of shape B is 312.5 cm².