Mathematics

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 2r, Therefore, the length, L, of arc XY is given as;

L 2r.

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 r²

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 +  2r

                                         = 2r +  2r  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 = absinθ = r²sinθ)

Area of Segment = area of sector – area of triangle

                               = ( r² – r²sinθ) square units.

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 120⁰ at the Centre.

• Find the perimeter of the sector of a circle of radius 7cm which subtends angle 150⁰ at the Centre.
• Find the length of the chord of a sector of a circle of radius 14cm which subtends angle 90⁰ at the center.
• Find the area of a sector of a circle of radius 7cm which subtends angle 60⁰ 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.