**WEEK: 1-2**

**OBJECTIVES:**

**The students should be able to;**

**Find the length of an arc****Find the area of a sector****Determine the perimeter of sector of a circle****Determine the area of segment of a circle****Determine the perimeter of a segment of a circle**

**ARC OF A CIRCLE**:

An arc of a circle is a portion of the circumference of a circle that is a cut-off from the part of a circle by two radii of the circle itself or by a chord of a circle.

In the figure below AB is an arc of circle center **O** cut-off by radii **OA** and **OB.**

A line joining two points on a circumference of the circle is a **Chord. **A chord divides the circle into two segments. Any chord that passes through the center of the circle is known as the **Diameter. **While** Circumference **is the distance around the circle.

**LENGTH OF A CIRCLE**

In determining of a length of a circle, two dimensions are necessary; these are

- The radius of the circle
- The angle subtended by the arc at the center

Arc **XY **subtends an angle of**. **The circumference of the circle is **2****r**, Therefore, the length, **L**, of arc **XY **is given as;

**L**** 2****r.**

Where is the ratio of angle subtended by the sector to the sum of angles in the circle.

**EXAMPLE 3.**

**SECTOR OF A CIRCLE**

A sector of a circle is a plain surface which is a part of the circle bounded by two radii and an arc cut-off by two radii.

**AREA OF A SECTOR OF A CIRCLE**

The area of a sector of a circle is found similar to the length of an arc of a circle. Given that a circle of radius **r **and the angle subtended at the center by the sector is**.**

Area of a sector will be = **L**** ****r ^{2}**

**EXAMPLES**

**PERIMETER OF A SECTOR**

The perimeter of a sector is the distance round edge of a circle. This distance comprises the two radii and the length of an arc contained in the sector.

In the figure above, the perimeter of the sector **AOB** will be sum of radii **AO, OB** and the arc **AB.**

Perimeter of a sector is: **r + r + arc AB**

** = r + r + **** 2****r**

** = 2r + **** 2****r or 2r****(1 + ****).**

**EXAMPLES**

** **

** **

** ****LENGTH OF A CHORD**

**(a)** Assuming the chord AB subtends an angle **Ө** at the center of the circle. **(b) **Let a perpendicular be dropped from **O** to **AB** at **E.** The figure below then becomes as shown below. The angle **Ө** has been bisected to become θ/2 each.

**|AB|= |AE| + |EB|,** but **|AE| = |EB|** because E is the midpoint of **AB.** Therefore **|AB| = 2|AE|.**

**ΔAEO** is a right-angled triangle, by trigonometric ratios

**AE/r = sinθ/2 **

Therefore, **AE = r****,** while

**AB = 2r ×sinθ/2 **** units**

**EXAMPLES**

**EXAMPLE 2**

**EXAMPLE 3**

**AREA OF A SEGMENT OF A CIRCLE**

In the above diagram the area of the segment **ABD** is calculated using the method, subtracting the area of triangle **OAB** from the area of the sector **OADB**.

**(Recall that area of a triangle is = ****absin****θ = ****r ^{2}sin**

**Area of Segment = area of sector – area of triangle**

= (** ****r ^{2 }– **

**EXAMPLES**

** **** **

**PERIMETER OF A SEGMENT OF A CIRCLE**

The perimeter of a segment is the distance round the segment. This distance consists of the chord and length of the arc forming the segment.

In the Segment of a circle above, the perimeter is the sum of the chord **AB** and the arc **AXB.**

Therefore perimeter of a segment will be:

**P = **** 2rsin**** ****or P = 2r(****+**** sin****) units.**

**EXAMPLES**

**EXAMPLE 3.**

**HOME ASSIGNMENT.**

**Find the length of an arc of the sector of a circle of radius 14cm which subtends angle 120**^{0 }at the**Centre.**

**Find the perimeter of the sector of a circle of radius 7cm which subtends angle 150**^{0}at the Centre.**Find the length of the chord of a sector of a circle of radius 14cm which subtends angle 90**^{0 }at the center.**Find the area of a sector of a circle of radius 7cm which subtends angle 60**^{0 }at the center.

**SEND YOUR SOLVING/ANSWERS TO THIS E-MAIL ADDRESS: This email address is being protected from spambots. You need JavaScript enabled to view it.**

**(07037022305)**

**PREPARED BY:**

**MADAM OBI ELIZABETH**

**BRIGHT L.L.C.E. EZEUKWU.**