5.1b Circles - Mathematics

5.1.2 Circumference of a Circle

$\begin{array}{l} circumference = π d, where d = d i a m e t e r \\ = 2 π r, where r = r a d i u s \\ π (p i) = \frac{22}{7} or 3.142 \end{array}$ $\begin{array}{l} circumference = π d, where d = d i a m e t e r \\ = 2 π r, where r = r a d i u s \\ π (p i) = \frac{22}{7} or 3.142 \end{array}$

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

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

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

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

Example:

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

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

5.1.4 Area of a Circle

$\begin{array}{l} Area of a circle = π \times {(radius)}^{2} \\ = π r^{2} \end{array}$ $\begin{array}{l} Area of a circle = π \times {(radius)}^{2} \\ = π r^{2} \end{array}$

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

Solution:
(a)

\begin{array}{l} Area of a circle = π r^{2} \\ = \frac{22}{7} \times 7 \times 7 \\ = 154 {cm}^{2} \end{array}

$\begin{array}{l} Area of a circle = π r^{2} \\ = \frac{22}{7} \times 7 \times 7 \\ = 154 {cm}^{2} \end{array}$

(b)

\begin{array}{l} Diameter of circle = 10 cm \\ Radius of circle = 5 cm \\ Area of circle = π r^{2} \\ = \frac{22}{7} \times 5 \times 5 \\ = 78.57 {cm}^{2} \end{array}

$\begin{array}{l} Diameter of circle = 10 cm \\ Radius of circle = 5 cm \\ Area of circle = π r^{2} \\ = \frac{22}{7} \times 5 \times 5 \\ = 78.57 {cm}^{2} \end{array}$

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

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

Example:

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

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