Final answer:
The relationship between the number of sides of a polygon and its area is complex, but for regular polygons, as the number of sides increases, the polygon resembles a circle more closely. The area of a square is proportional to the square of its side length, so if side length doubles, the area quadruples.
Step-by-step explanation:
The relationship between the number of sides of a polygon and its area can be complex, depending on the shape and dimensions of the polygon.
However, in the case of regular polygons (where all sides and angles are equal), as the number of sides increases, the figure more closely approximates a circle, which has the maximum area for a given perimeter. Considering squares specifically, the area of a square is directly proportional to the square of its side length.
For example, let's consider Marta's squares. The smaller square has a side length of 4 inches, so its area is 4 inches × 4 inches = 16 square inches. Marta's larger square has side lengths that are twice the first square, which means the side lengths are 8 inches.
The area of the larger square is 8 inches × 8 inches = 64 square inches, which is exactly four times bigger than the area of the smaller square (since 8^2 = (2 × 4)^2 = 4^2 × 2^2). This illustrates that the area of a square increases by the square of the factor by which its side length is increased.
Where A is the area, s is the length of each side, and n is the number of sides of the polygon.
For example, if you have a square with side length 's', the area would be A = s2. If you have a pentagon with side length 's', the area can be calculated using the formula A = (s2 * 5) / 4 * tan(π / 5).