# 10.1 Circles I

     
   
10.1 Circles I
   

   
10.1.1 Parts of a Circle
   
1. A circle is set of points in a plane equidistant from a fixed point.
   

   2.
Parts of a circle:
   
(a)    The centre, O, of a circle is a fixed point which is equidistant from all points on the circle.
   
   
   
   
   
   
 
   

   
   (b)
A sector is the region enclosed by two radii and an arc.
   
   
   
   
   
 
     

   
   (c)
An arc is a part of the circumference of a circle.
   
   
   

   
   (d)
A segment is an area enclosed by an arc and a chord.
   

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

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

   

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

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

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

   
   
 
 
     
   
10.1.4 Area of a Circle
   
   
   
   
         
   

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

    Solution
:
   
(a)
   
    $\begin{array}{l}\text{Area of a circle}=\pi {r}^{2}\\ \text{}=\frac{22}{\overline{)7}}×\overline{)7}×7\\ \text{}=154{\text{cm}}^{2}\end{array}$    
 
     
(b)
   
    $\begin{array}{l}\text{Diameter of circle}=10\text{cm}\\ \text{Radius of circle}=5\text{cm}\\ \text{Area of circle}=\pi {r}^{2}\\ \text{}=\frac{22}{7}×5×5\\ \text{}=78.57{\text{cm}}^{2}\end{array}$

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

    Example
:
   
   

   
   

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