The area \( A \) of a circle can be calculated using the formula:

\[
A = \pi r^2
\]

**Where:**
– \( r \) is the radius of the circle.
– \( \pi \) (pi) is a mathematical constant approximately equal to 3.14159.

### Key Points:
1. **Radius vs. Diameter:** Ensure you use the **radius** (distance from the center to the edge). If given the diameter \( d \), first convert it to radius: \( r = \frac{d}{2} \).

2. **Derivation Insights:**
– **Integration:** The area can be derived by integrating the function \( y = \sqrt{r^2 – x^2} \) over \([-r, r]\) and exploiting symmetry.
– **Geometric Approximation:** Archimedes’ method of exhaustion using polygons, or summing infinitesimal circular rings (each with area \( 2\pi r \, dr \)).

3. **Units:** The area will be in square units (e.g., \( \text{m}^2 \)) if the radius is in linear units (e.g., meters).

4. **Applications:** Used in fields ranging from engineering to astronomy for calculating circular areas, volumes of cylinders, and more.

**Example:**
For a circle with radius \( r = 5 \, \text{cm} \):
\[
A = \pi (5)^2 = 25\pi \, \text{cm}^2 \approx 78.54 \, \text{cm}^2
\]