The formula for the area of a circle, \( A = \pi r^2 \), where \( r \)
is the radius, is derived from the basic properties of circles and
fundamental principles of geometry.
Here's why \( \pi r^2 \) represents the area of a circle:
1.
Definition of a Circle:
A circle is a set of all points in a plane that are equidistant from a
fixed center. The distance from the center to any point on the circle is
known as the radius.
2.
Understanding the Formula:
\( \pi r^2 \) is an expression that represents the area of a circle,
where:
 \( r \) is the radius of the circle.

\( \pi \) is a mathematical constant, approximately equal to
3.14159, which relates the circumference and diameter of a
circle.
3.
Derivation of the Formula:
The formula is derived from the understanding that the area of a circle
can be calculated by "slicing" the circle into infinitesimally small
sectors (like a pizza or pie) and then rearranging them to form a shape
very close to a parallelogram. This allows us to use familiar geometry
to find the circle's area.
When the circle is rearranged into a parallelogram, the base of this
parallelogram is the circumference of the circle (which is \( 2\pi r
\)), and the height is the radius (\( r \)).
The area of a parallelogram is given by the formula: \(\text{Area} =
\text{base} \times \text{height}\).
In this case, the base is the circumference (\( 2\pi r \)) and the
height is the radius (\( r \)):
\[ \text{Area of parallelogram} = 2\pi r \times r = 2\pi r^2 \] However,
this parallelogram formed is actually half of the area of the circle.
Hence, the area of the circle, which corresponds to the formula \( \pi
r^2 \), is half of the area of this parallelogram:
\[ \text{Area of circle} = \frac{1}{2} \times \text{Area of} \\\\\
\text{ parallelogram} = \frac{1}{2} \times 2\pi r^2 = \pi r^2 \]
Therefore, \( \pi r^2 \) represents the area of a circle and is derived
from geometric principles and mathematical reasoning, specifically from
breaking down the circle into infinitesimal sectors and rearranging them
to form a shape whose area can be calculated using known geometric
formulas.