The area of a circle is calculated using the formula: $$A = \pi r^2$$
where $r$ is the radius (half the diameter). Make sure you use the radius, not the diameter.
Example: Find the area of a circle with diameter 10 cm. $$r = 5\text{ cm}, \quad A = \pi \times 5^2 = 25\pi \approx 78.5\text{ cm}^2$$
Finding the radius from the area: rearrange the formula. $$r = \sqrt{\frac{A}{\pi}}$$
Area of a semicircle: $\frac{1}{2}\pi r^2$ — then add the area of any attached shapes if needed.
Composite shapes: for shapes involving circles and rectangles/triangles, calculate each area separately and add or subtract.
$$\text{Area of annulus (ring)} = \pi R^2 - \pi r^2 = \pi(R^2 - r^2)$$
Common error: squaring the diameter instead of the radius ($\pi d^2$ instead of $\pi r^2$), or forgetting to halve the diameter to get the radius.