Area and Circumference of Circles

To work with circles, you need to know three important parts:

• Radius ($r$): The distance from the center to the edge.
• Diameter ($d$): The distance across the circle through the center. The diameter is always twice the radius ($d = 2r$).
• Pi ($\pi$): A special mathematical constant representing the ratio of a circle's circumference to its diameter. We usually approximate $\pi \approx 3.14$ or $\frac{22}{7}$.

Circumference: The Distance Around

The circumference is the perimeter, or the distance around the outside edge of the circle.

Formulas: $C = \pi d$ or $C = 2 \pi r$

Example: Find the circumference of a circle with a diameter of 14. Since $d = 14$, use the formula $C = \pi d$: $C = 14\pi \approx 14 \times 3.14 = 43.96$

Area: The Space Inside

The area measures the total flat space inside the circle.

Formula: $A = \pi r^2$ (Remember: Always square the radius first, then multiply by $\pi$!)

Example: Find the area of a circle with a radius of 5. $A = \pi (5)^2$ $A = 25\pi \approx 25 \times 3.14 = 78.5$

Semicircles (Half Circles)

A semicircle is exactly half of a circle.

Area of a Semicircle: Just find the area of a full circle and divide by 2. $A = \frac{1}{2} \pi r^2$

Perimeter of a Semicircle: The perimeter includes the curved edge (which is half the circumference) plus the straight bottom edge (which is the diameter). $P = \pi r + d$

Example: Find the area and perimeter of a semicircle with a radius of 6.

• Area: $A = \frac{1}{2} \pi (6^2) = \frac{1}{2} \pi (36) = 18\pi \approx 56.52$
• Perimeter: The diameter is $2 \times 6 = 12$. The curved part is $\pi \times 6 = 6\pi$. $P = 6\pi + 12 \approx (6 \times 3.14) + 12 = 18.84 + 12 = 30.84$