formula_xm f2

8/8/2019 formula_xm f2

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MATHEMATICAL FORMULAE

RELATIONS

Distance = ( ) ( )2 2

2 1 2 1 x x y y +

Midpoint, ( ) 1 2 1 2, = ,2 2

x x y yx y

+ +

Phythagoras Theorem = 2 2 2c a b= +

SHAPE AND SPACE

1. Area of rectangle = length width

2. Area of triangle =1

2 base height

3. Area of parallelogram = base height

4. Area of trapezium =1

2 (sum of parallel sides) height

5. Circumference of circle = d = 2 r

6. Area of circle = 2r

7. Curved surface area of cylinder = 2 r h

8. Surface area of sphere = 4 2r

9. Volume of right prism = cross sectional area length

10. Volume of cuboid = length width height

11. Volume of cylinder = 2r h

12. Volume of cone =21

3r h

13. Volume of sphere =34

3r

14. Volume of right pyramid =1

3

base area height

15. Sum of interior angles of a polygon = ( )2 180n

16.arc length angle substended at centre

circumference of circle 360=

17.area of sector angle substended at centre

area of circle 360

=


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CIRCLES

Identifying a circle- we have learnt a point that moves at a constant distant from a fixed point is a

circle- a circle is a set of point that is equidistant from the fixed point


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Circumference of a circle

- the ratio of the circumference to diameter is a constant- this constant is known as pi and is represented by the symbol

-22

7 = OR 3.142 =

Circumference = d OR

Circumference = 2 r , r = radius

- Examples:

1) Calculate the circumference of a circle with a diameter of 7 cm ( use22

7 = )

2) Calculate the circumference of a circle with a radius 3.2 cm ( use 3.142 = )

3) Find the diameter of a circle which has a circumference of 44 cm. ( use22

7 = )

4) If a circle has a circumference of 66 cm, find its radius. ( use22

7 = )

5. The tyre of a bicycle rotates 50 times per minutes. Given the diameter of the tyre is 42 cm, find the


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distance traveled by the bicycle in 5 minutes. Give your answer in meters.

6. A piece of wire in the shape of a rectangle 24 cm long and 20 cm wide is reshaped into a circle. Find the

radius of the circle. ( use22

7 = )

Arc of a circle

length of arc angle at centre

circumference 360

=

Examples :

1) Calculate the length of an arc that subtends an angle of72 at the centre of a circle with a radius

of 21 cm

2) O

xAB


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Find the value ofx given that the circle with centre O has a radius of 14 m and the minor arcAB is

44 m long.

3) The length of an arc of a circle of radius

1

10 2 cm is 22cm. Find the angle at the centre of the sector

4) In the figure, the length of the minor arc PQ is

1

13 5 cm and angle POQ is 84

. Calculate the

a) radius of the circleb) length of the major arc

5) The figure shows a right-angled triangle joined to a semicircle. find the perimeter of the figure.


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6) The figure given shows a rectanglePQRSoverlapped by two quadrantsPAD andRCB of radius

7cm. IfPS= 10cm andPQ = 14cm, calculate the perimeter of the shaded area.

Area of a circle

P A S

B

RC

Q

D

6cm

8cm


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where ris the radius

- Examples:

1) Calculate the area of a circle with a radius = 1.4 cm. ( Take22

7 = )

2) Calculate the area of a circle with a diameter. ( Take 3.142 = )

3) The area of a circle is 2154cm . Find its radius and diameter ( Use22

7 = )

4) Find the area of the circle with the circumference of 88cm. ( Use22

7

= )

5) The figure shows two concentric circles with centre O. Find the area of the annulus.

Area of circle = r2

O

3 cm4 cm


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6) Take =7

22, find the area of the shaded regions

Area of a Sector of a Circle

Important!

ConcentricCircles= circles having the same centre

Annulus= area between two concentriccircles= area of big circle area of small

O P

42cm

=

360CentreAtAngle

CircleofAreaSectorofArea


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OR

Examples:

1) Find the area of the shaded region. ( Take22

7 = )

2) Given that the area of the shaded sector of circle with radius of 6.3 cm is 27.72 2cm , calculate the angle

substended at centre of the circle. ( Use22

7 = )

3) The area of the shaded region in the circle with centre O is 48 2cm . Given that the angle substended at

2Area of sector360

r

=

O

72o

7cm


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the centre is 144 , calculate the radius of the circle. ( Use22

7 = )

4) The diagram below shows a circle with centre O and a radius of 14mm. OAB is a right-angled triangle.

Find the area of the shaded part.

5) In the diagram as shown, PS and QR are the arcs of two circles with centre O. Find the area of theshaded region. Give your answer in

PMR Past Year Questions (Chapter 10 Circles)

O

A

B


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2004 Paper 1 Question 19Diagram 12 shows a circle with centre O.

The radius of the circle is2

110 cm. Calculate the length, in cm, of the minor arc MN. (Use

7

22= )

A2

55B

2

77C

8

1155D

8

1617

2004 Paper 1 Question 20

In Diagram 13,POR is the diameter of circlePQR.

Given thatPQ = QR andPR = 14cm, calculate the area, in cm2, of the shaded region. (Use7

22= )

A 56 B 93 C 105 D 142

2005 Paper 1 Question 40Diagram 25 shows a circle with centre O. The length of the minor arcPQ is 3.3 cm.

Calculate the radius, in cm, of the circle. (Use7

22= )

A 2.42 B 4.50 C 14.85 D 29.70


Diagram 15 shows a circle with centre O.


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Calculate the length, in cm, of the minor arcPQ. (Use7

22= )

A 22 B 44 C 88 D 176

2006 Paper 1 Question 22Diagram 17 shows two sectors, QSR and UST, with the common centre S.RSUand QSTare straight lines.

It is given that ST= 2QS. Calculate the area, in cm2, of the sectorUST. (Use7

22= )

A 11 B 19.25 C 38.50 D 77

2007 Paper 1 Question 23Diagram 16 shows a circle with centre O.PT, QU,RVand SWare diameters of the circle.

Which of the following minor arcs is the longest?

A TU B UV C WP D RS

2007 Paper 1 Question 24Diagram 17 shows a circle with centre O and radius 6 cm.


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Calculate the area, in cm2, of the shaded region.

A 18 B 5.4 C 10.2 D 30.6

2007 Paper 1 Question 31Diagram 22 shows a square OPQR and an arcRSPwith centre O.

Calculate the perimeter, in cm, of the whole diagram.A 61 B 58 C 47 D 33


Diagram 12 shows a circle with centre O.

Calculate the area, in cm2, of the coloured region. (Use7

22= )

A 105 B 116 C 539 D 616


Diagram 13, O is the centre of the circle andPOR is a diameter of the circle.PQRSis a rectangle.


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It is given thatPR = 10 cm andPQ = 6 cm. Find the area, in cm2, of the coloured region.

A 25 48 B 25 96 C 100 48 D 100 96


In Diagram 19,PQRSis a square and STUis an arc of a circle with centreP.

The area ofPQRSis 576 cm2. Calculate the length, in cm, of the arc STU.

A 16 B 28 C 32 D 48

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