**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.

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)

(b)

**A**

**sector**is the region enclosed by two radii and an arc.

(c)

(c)

**An**

**arc**is a part of the circumference of a circle.

(d)

(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. $\left(\pi =\frac{22}{7}\right)$

SolutionSolution

**:**

**10.1.3 Arc of a Circle**

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

$$\overline{)\begin{array}{l}\text{}\\ \text{}\frac{\text{Lengthofarc}}{\text{Circumference}}=\frac{\text{Angleatcentre}}{{360}^{o}}\text{}\\ \text{}\end{array}}$$

*Example***:**

Calculate the length of the minor arc

*AB*of the circle above. $\left(\pi =\frac{22}{7}\right)$

SolutionSolution

**:**

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

**10.1.4 Area of a Circle**

$$\overline{)\begin{array}{l}\text{Areaofacircle}=\text{}\pi \times {\left(\text{radius}\right)}^{2}\text{}\\ \text{}=\pi {r}^{2}\end{array}}$$

*Example***:**

Calculate the area of each of the following circles that has

(a) a radius of 7 cm,

(b) a diameter of 10 cm.

$\left(\pi =\frac{22}{7}\right)$

SolutionSolution

**:**

(a)

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

(b)

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

**10.1.5 Area of a Sector**

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

$$\overline{)\begin{array}{l}\text{}\\ \text{}\frac{\text{Areaofsector}}{\text{Areaofcircle}}=\frac{\text{Angleatcentre}}{{360}^{o}}\text{}\\ \text{}\end{array}}$$

ExampleExample

**:**

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