How should you manipulate the formula for the area of a circle to express A?

The area of a circle is derived from its radius. The formula is based on the relationship between the radius and the area, incorporating the mathematical constant π (pi), which is essential for calculations involving circles. The standard geometric formula for the area 'A' of a circle is given by A = πR², where R represents the radius.

In this formula, squaring the radius (R²) accounts for the two-dimensional nature of the area. It effectively captures the space enclosed by the circle as the radius expands. Therefore, when expressing the area of a circle, the term πR² accurately represents the area based on its radius, providing a clear and direct correlation in geometry.

The other options do not align with the geometric definition of area in relation to a circle. For instance, expressing the area as πR or R² / π does not conform to the established mathematical principles regarding circles, failing to include the necessary squared term of the radius. Similarly, taking the square root of R divided by π does not grasp the intended geometric relationship, thus reinforcing that the correct representation of the area is indeed A = πR².