Understanding the Area of a Semicircle: Formulas and Calculations

Semicircle Definition and Basic Formula

A semicircle is defined as half of a full circle. The area of a full circle is given by the formula (A pi r^2), where (r) is the radius of the circle. Since a semicircle is half of a circle, the formula for the area of a semicircle can be derived by taking half of the area of the full circle:

Area of a Semicircle with Radius (r)

The area of a semicircle with radius (r) is given by the formula:

(A frac{pi r^2}{2})

Area of a Semicircle with Diameter (d)

Since the diameter (d) of a circle is twice the radius ((d 2r)), we can substitute (r frac{d}{2}) into the formula to find the area of a semicircle with diameter (d):

(A frac{pi (frac{d}{2})^2}{2} frac{pi d^2}{8})

Steps to Find the Area of a Semicircle

Identify the radius or diameter of the semicircle. If you have the radius, use the formula (frac{pi r^2}{2}). If you have the diameter, use the formula (frac{pi d^2}{8}). Plug in the given value into the appropriate formula and calculate the area.

Example Calculation:

Suppose you have a semicircle with diameter 6 cm. To find its area:

(d 6) (A frac{pi d^2}{8} frac{pi times 6^2}{8} frac{36pi}{8} 4.5pi approx 14.14 text{ square centimeters})

This step-by-step approach and the provided formulas ensure accurate calculation of the area of a semicircle, whether the given values are in terms of radius or diameter.