Circles (circumference, area) Knowledge-Based Study

Study notes

1. Basic Elements of a Circle

   A circle is the set of all points in a plane that are at a fixed distance from a given point called the center.

   • Radius ($r$): The distance from the center of the circle to any point on its boundary.
   • Diameter ($d$): The distance across the circle passing through the center. The diameter is twice the length of the radius: $$d = 2r$$

   The Constant Pi ($\pi$)

   For any circle, the ratio of its perimeter (circumference) to its diameter is always constant. This ratio is represented by the Greek letter $\pi$ (pi):

   $$\pi = \frac{\text{Circumference}}{\text{Diameter}} \approx 3.14159\dots$$

   In school mathematics, $\pi$ is commonly approximated as $3.14$ or as the fraction $\frac{22}{7}$, unless specified otherwise.

   What is Circumference?

   The circumference ($C$) is the linear distance around the outside edge of a circle. It represents the perimeter of the circle.

   Circumference Formulas

   Since $\pi = \frac{C}{d}$, rearranging the formula gives:

   $$C = \pi d$$

   Substituting $d = 2r$ gives the most common form:

   $$C = 2\pi r$$

   Worked Example

   Problem: Find the circumference of a circle with a radius of 7 cm. (Use $\pi = \frac{22}{7}$)

   Solution:

   1. Identify the given values: $r = 7\text{ cm}$.
   2. Apply the circumference formula: $$C = 2\pi r = 2 \times \frac{22}{7} \times 7 = 44\text{ cm}$$

   The circumference of the circle is 44 cm.

   What is the Area of a Circle?

   The area ($A$) of a circle is the total two-dimensional space enclosed within its boundary.

   Area Formula

   The area of a circle with radius $r$ is given by:

   $$A = \pi r^2$$

   Note: The radius is squared first, and then multiplied by $\pi$. The units of area are always square units, such as $\text{cm}^2$ or $\text{m}^2$.

   Worked Example

   Problem: A circular rug has a diameter of 6 m. Calculate its area in terms of $\pi$ and to two decimal places using $\pi \approx 3.14$.

   Solution:

   1. Find the radius from the diameter: $$r = \frac{d}{2} = \frac{6}{2} = 3\text{ m}$$
   2. Calculate the exact area in terms of $\pi$: $$A = \pi r^2 = \pi \times 3^2 = 9\pi\text{ m}^2$$
   3. Approximate using $\pi \approx 3.14$: $$A \approx 9 \times 3.14 = 28.26\text{ m}^2$$

   Finding the Radius from Known Measures

   When given the circumference or area, you can solve for the radius or diameter algebraically:

   • From Circumference: $$r = \frac{C}{2\pi}$$
   • From Area: $$r = \sqrt{\frac{A}{\pi}}$$

   Semicircles

   A semicircle is half of a circle:

   • Area of a Semicircle: Exactly half of the full circle's area: $$A_{\text{semi}} = \frac{1}{2}\pi r^2$$
   • Perimeter of a Semicircle: The curved boundary plus the straight diameter: $$P_{\text{semi}} = \frac{1}{2}(2\pi r) + d = \pi r + 2r$$

   Worked Example

   Problem: The area of a circle is $100\pi\text{ cm}^2$. What is its circumference?

   Solution:

   1. Set up the area equation to find $r$: $$\pi r^2 = 100\pi \implies r^2 = 100 \implies r = 10\text{ cm}$$
   2. Substitute $r = 10\text{ cm}$ into the circumference formula: $$C = 2\pi r = 2\pi(10) = 20\pi\text{ cm}$$