5.1b Circles

5.1.2 Circumference of a Circle

$\boxed{\begin{array}{l}\text{ circumference}=\pi d,\text{ where }d=diameter\\ =2\pi r,\text{ where }r=radius\\ \\ \pi (pi)=\frac{22}{7}\text{ or }3.142\end{array}}$

Example:
Calculate the circumference of a circle with a diameter of 14 cm. $\left(\pi =\frac{22}{7}\right)$

Solution:
$\begin{array}{l}\text{Circumference}=\pi \times \text{ Diameter}\\ =\frac{22}{7}\times 14\\ =44\text{ cm}\end{array}$

5.1.3 Arc of a Circle
The length of an arc of a circle is proportional to the angle at the centre.

$\boxed{\begin{array}{l}\\ \frac{\text{Length of arc}}{\text{Circumference}}=\frac{\text{Angle at centre}}{{360}^o}\\ \end{array}}$

Example:


Calculate the length of the minor arc AB of the circle above.
$\left(\pi =\frac{22}{7}\right)$

Solution:
$\begin{array}{l}\frac{\text{Length of arc}}{\text{Circumference}}=\frac{\text{Angle at centre}}{{360}^o}\\ \\ \text{Length of arc }AB=\frac{{120}^o}{{360}^o}\times 2\times \frac{22}{7}\times 7\\ =14\frac{2}{3}\text{ cm}\end{array}$

5.1.4 Area of a Circle

$\boxed{\begin{array}{l}\text{ Area of a circle }=\pi \times {\left(\text{radius}\right)}^2\\ =\pi r^2\end{array}}$

Example:
Calculate the area of each of the following circles that has
(a) a radius of 7 cm,
(b) a diameter of 10 cm.
$\left(\pi =\frac{22}{7}\right)$

5.1.5 Area of a Sector
The area of a sector of a circle is proportional to the angle at the centre.

$\boxed{\begin{array}{l}\\ \frac{\text{Area of sector}}{\text{Area of circle}}=\frac{\text{Angle at centre}}{{360}^o}\\ \end{array}}$

Example:


$\begin{array}{l}\text{Area of sector }ABC\\ =\frac{{72}^o}{{360}^o}\times \frac{22}{7}\times 7\times 7\\ =30\frac{4}{5}{\text{ cm}}^2\end{array}$