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MATHEMATICS - FORM 2 CIRCLES

PARTS OF A CIRCLE (A) Identifying a circle as a set of points equidistant from a fixed point A circle is a locus with all the points on the plane at a constant distance from a fixed point (known as the centre).

(B) Identifying parts of a circle The parts of a circle:

(a) Centre - fixed point in the middle of the circle with a constant distance from all points on the circle. (b) Circumference - the length of the border of the circle. (c) Radius - the length of straight line from the centre to any point on the circumference. (d) Diameter- the length of a straight line joining any two points on the circumference passing through the centre. The length of the diameteris twice the radius. (e) Chord - a straight line joining any two points on the circumference. The diameter is the longest chord in a circle. (f) Arc - a part of the circumference of a circle with end-points on the circle.

(g) Sector - the region bounded by an arc and two radii

(h) Segment - the region bounded by a chord and an arc

(i) Semicircle - one half of a circle formed within an arc and a diameter

(j) Quadrant - one quarter of a circle formed by an arc and two perpendicular radii.

(C) Identifying parts of a circle A circle or parts of a circle can be drawn using a straight edge, a pair of compasses or a protactor.

Draw a circle Draw a circle with centre O and radius 1 cm.

(a) Mark a point and label it as O. (b) Open a pair of compasses to a length of 1 cm. (c)

Place the pointed end of the compasses at O and draw the circle.

Draw a diameter Draw a diameter of length 3 cm passing through a point R in a circle with centre O.

(a) Draw a circle i=with centre O and a radius of 1.5 cm. (b) Mark a point R in the circle. (c) Using a ruler, join O to R and extend both ends to reach the circumference of the circle.

Draw a chor Construct a circle with radius 1.5 cm. Then draw a chord with a length of 2 cm which passes through P on the circumference.

(a) Draw a circle with a radius of 1.5 cm. (b) Mark a point P on the circumference. (c) Open the compasses to a length of 2 cm. (d) Place the pointed end of the compasses at P and mark an arc intersecting the circumference.

(e) Using a ruler, joint the two points.

Draw a sector Draw the sector of a circle given that the angle at the centre is 80° and its radius is 1.5 cm.

(a) Draw a circle of radius 1.5 cm with centre O. (b) Draw a radius and name it OP. (c) Using a protractor, draw an angle POQ = 80°. (d) Using a ruler, join O to Q to obtain the sector. POQ is a sector of the circle with an angle of 80° at the centre and a radius of 1.5 cm.

(D) Determining the centre and radius of a circle by construction The perpendicular bisectors of two unparallel chords of a circle intersect at the centre of the circle.

Determine the centre and radius of the circle given.

(a) Draw two unparallel chords PQ and RS in the circle.

(b) Construct the perpendicular bisectors of both chords. (c) The intersection point of the perpendicular bisectors of the chords is the centre, O, of the circle. (d) Measure the length of OP, OQ, OR or OS to get the radius of the circle.

CIRCUMFERENCE OF A CIRCLE (A) The value of

π

(B) Finding the circumference of a Circle

Calculate the circumference of a circle with a (a) diameter of 7 cm. (b) radius of 14 cm.

(C) Finding the Diameter of a Circle

If the circumference of a circle is 44 cm, find its diameter.

(D) Finding the Radius of a Circle

If a circle has a circumference of 66 cm, find its radius.

(E) Solving Problem Involving the Circumference of a Circle

The above figure shows a piece of paper, ABCD, in the shape of a square. The shaded part consist of four quadrants with radius 4 cm. Calculate the perimeter f the shaded part.

Length of a side of the square ABCD = 4 cm + 4 cm = 8 cm Perimeter of ABCD = 4 + 8 = 32 cm

Perimeter of the shaded part = Perimeter of ABCD = 4 x 8 = 32 cm + 25.136 cm = 57.136 cm Therefore, the perimeter of the shaded part is 57.136 cm.

ARC OF A CIRCLE (A) Deriving the formula for the length of an arc 1. An arc of a circle is any part of the curve that makes the circle. 2. The length of an arc is proportional to the angle formed by the arc at the centre of the circle.

(B) Finding the length of an arc given the angle at the centre and the radius

Find the length of arc which subtends an angle of 60° at the centre of a circle of radius 21 cm.

Length of arc

(C) Finding the angle at the centre given the length of the arc and the radius of the circle

In the diagram, O is the centre of the circle. Find the value of x.

(D) Finding the radius of a circle given the length of the arc and the angle at the centre

The diagram shows a circle with centre O. Find the radius of the circle.

The radius of the circle is 9 cm.

(E) Solving problems involving arcs of a circle

The diameter of a circular shaped pie is 21 cm. It is divided into several equal slices with the arc of each slice being 8.25 cm. What is the angle at the centre of each slice of the pie?

The angle at the centre of each slice of the pie is 45°.

AREA OF A CIRCLE (A) Finding the Area of a Circle The area of a circle is the area of the reqion bounded by the circumference.

(B) Finding the Radius and the Diameter of a Circle

Find the area of the circle is 154². Find its radius and diameter.

(C) Finding the Area of a Circle Given the Circumference Find the area of the circle with the circumference of 88 cm.

Therefore, the area of the circle is 616 cm².

(D) Solving Problems Involving Areas of Circles

The above figure shows two circles with centre O and with a radius of 9 cm and 5 cm respectively. Find the area of the shaded part.

AREA OF A SECTOR OF A CIRCLE (A) Deriving the formula of the area of a sector 1. The area of a sector is the area enclosed between an arc and the two radii at either end of the arc. 2. The area of a sector is proportional to the angle at the centre of the circle.

(B) Finding the area of a sector given the radius and angle at the centre

Find the area of shaded sector above where O is the centre of the circle.

(C) Finding the angle at the centre given the radius and area of a sector.

(D) Finding the radius given the area of a sector and the angle at the centre

In the diagram, the area of the major sector POQ is 702.24 cm². Find the radius of the circle.

The radius of the circle is 16.8 cm.

(E) Solving problems involving the area of sectors and area of circles

Richard drew a semicircle with centre O on a piece of rectangular paper PQRS. He only used the region formed by the sector with an angle of 126°. Calculate the remaining area of the paper.

The above figure shows two circles with centre O. The straight line AEOGC is perpendicular to the straight line BFOHD, and OE = AF = 6 cm. Find the area of the shaded part.

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PARTS OF A CIRCLE (A) Identifying a circle as a set of points equidistant from a fixed point A circle is a locus with all the points on the plane at a constant distance from a fixed point (known as the centre).

(B) Identifying parts of a circle The parts of a circle:

(a) Centre - fixed point in the middle of the circle with a constant distance from all points on the circle. (b) Circumference - the length of the border of the circle. (c) Radius - the length of straight line from the centre to any point on the circumference. (d) Diameter- the length of a straight line joining any two points on the circumference passing through the centre. The length of the diameteris twice the radius. (e) Chord - a straight line joining any two points on the circumference. The diameter is the longest chord in a circle. (f) Arc - a part of the circumference of a circle with end-points on the circle.

(g) Sector - the region bounded by an arc and two radii

(h) Segment - the region bounded by a chord and an arc

(i) Semicircle - one half of a circle formed within an arc and a diameter

(j) Quadrant - one quarter of a circle formed by an arc and two perpendicular radii.

(C) Identifying parts of a circle A circle or parts of a circle can be drawn using a straight edge, a pair of compasses or a protactor.

Draw a circle Draw a circle with centre O and radius 1 cm.

(a) Mark a point and label it as O. (b) Open a pair of compasses to a length of 1 cm. (c)

Place the pointed end of the compasses at O and draw the circle.

Draw a diameter Draw a diameter of length 3 cm passing through a point R in a circle with centre O.

(a) Draw a circle i=with centre O and a radius of 1.5 cm. (b) Mark a point R in the circle. (c) Using a ruler, join O to R and extend both ends to reach the circumference of the circle.

Draw a chor Construct a circle with radius 1.5 cm. Then draw a chord with a length of 2 cm which passes through P on the circumference.

(a) Draw a circle with a radius of 1.5 cm. (b) Mark a point P on the circumference. (c) Open the compasses to a length of 2 cm. (d) Place the pointed end of the compasses at P and mark an arc intersecting the circumference.

(e) Using a ruler, joint the two points.

Draw a sector Draw the sector of a circle given that the angle at the centre is 80° and its radius is 1.5 cm.

(a) Draw a circle of radius 1.5 cm with centre O. (b) Draw a radius and name it OP. (c) Using a protractor, draw an angle POQ = 80°. (d) Using a ruler, join O to Q to obtain the sector. POQ is a sector of the circle with an angle of 80° at the centre and a radius of 1.5 cm.

(D) Determining the centre and radius of a circle by construction The perpendicular bisectors of two unparallel chords of a circle intersect at the centre of the circle.

Determine the centre and radius of the circle given.

(a) Draw two unparallel chords PQ and RS in the circle.

(b) Construct the perpendicular bisectors of both chords. (c) The intersection point of the perpendicular bisectors of the chords is the centre, O, of the circle. (d) Measure the length of OP, OQ, OR or OS to get the radius of the circle.

CIRCUMFERENCE OF A CIRCLE (A) The value of

π

(B) Finding the circumference of a Circle

Calculate the circumference of a circle with a (a) diameter of 7 cm. (b) radius of 14 cm.

(C) Finding the Diameter of a Circle

If the circumference of a circle is 44 cm, find its diameter.

(D) Finding the Radius of a Circle

If a circle has a circumference of 66 cm, find its radius.

(E) Solving Problem Involving the Circumference of a Circle

The above figure shows a piece of paper, ABCD, in the shape of a square. The shaded part consist of four quadrants with radius 4 cm. Calculate the perimeter f the shaded part.

Length of a side of the square ABCD = 4 cm + 4 cm = 8 cm Perimeter of ABCD = 4 + 8 = 32 cm

Perimeter of the shaded part = Perimeter of ABCD = 4 x 8 = 32 cm + 25.136 cm = 57.136 cm Therefore, the perimeter of the shaded part is 57.136 cm.

ARC OF A CIRCLE (A) Deriving the formula for the length of an arc 1. An arc of a circle is any part of the curve that makes the circle. 2. The length of an arc is proportional to the angle formed by the arc at the centre of the circle.

(B) Finding the length of an arc given the angle at the centre and the radius

Find the length of arc which subtends an angle of 60° at the centre of a circle of radius 21 cm.

Length of arc

(C) Finding the angle at the centre given the length of the arc and the radius of the circle

In the diagram, O is the centre of the circle. Find the value of x.

(D) Finding the radius of a circle given the length of the arc and the angle at the centre

The diagram shows a circle with centre O. Find the radius of the circle.

The radius of the circle is 9 cm.

(E) Solving problems involving arcs of a circle

The diameter of a circular shaped pie is 21 cm. It is divided into several equal slices with the arc of each slice being 8.25 cm. What is the angle at the centre of each slice of the pie?

The angle at the centre of each slice of the pie is 45°.

AREA OF A CIRCLE (A) Finding the Area of a Circle The area of a circle is the area of the reqion bounded by the circumference.

(B) Finding the Radius and the Diameter of a Circle

Find the area of the circle is 154². Find its radius and diameter.

(C) Finding the Area of a Circle Given the Circumference Find the area of the circle with the circumference of 88 cm.

Therefore, the area of the circle is 616 cm².

(D) Solving Problems Involving Areas of Circles

The above figure shows two circles with centre O and with a radius of 9 cm and 5 cm respectively. Find the area of the shaded part.

AREA OF A SECTOR OF A CIRCLE (A) Deriving the formula of the area of a sector 1. The area of a sector is the area enclosed between an arc and the two radii at either end of the arc. 2. The area of a sector is proportional to the angle at the centre of the circle.

(B) Finding the area of a sector given the radius and angle at the centre

Find the area of shaded sector above where O is the centre of the circle.

(C) Finding the angle at the centre given the radius and area of a sector.

(D) Finding the radius given the area of a sector and the angle at the centre

In the diagram, the area of the major sector POQ is 702.24 cm². Find the radius of the circle.

The radius of the circle is 16.8 cm.

(E) Solving problems involving the area of sectors and area of circles

Richard drew a semicircle with centre O on a piece of rectangular paper PQRS. He only used the region formed by the sector with an angle of 126°. Calculate the remaining area of the paper.

The above figure shows two circles with centre O. The straight line AEOGC is perpendicular to the straight line BFOHD, and OE = AF = 6 cm. Find the area of the shaded part.

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