Circumference of a circle 17

By studying this chapter you will be able to acquire the following competencies.

Understanding different methods of measuring the circumference and the diameter of a circle.Discovering a relationship between the circumference and the diameter of a circle. Using the above relationship to calculate either the radius or the circumference of a circle when one out of the two is given.Using this relationship in day today life situations.

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« The locus of a point moving at a constant distance from a fixed point is a circle.

Dulani -Fathima -Suranga -Fathima -Dulani -

Its beautiful Fathima! You are wearing many bangles today.Yes, my uncle bought them for me.Fathima, do you know the length of the wire used to make one bangle?Sorry, I do not know.A bangle is circular in shape. The length of the wire used to make the bangle is the length around the bangle. That is its perimeter.

SANJEEWAYOGHURT

A bangle A Rs 1$- coin A Rs 2$- coin A cart wheel

A vehicle wheel A lid of a yoghurt cup

A C.D. A face of a clock

17.1 The circumference and the diameter of a circle.

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Suranga -

Dulani -

Suranga -

Rajen -

Dulani

Rajen -

Suranga -

Yes, but the length around a circle is called its circumference. Do you know the methods of measuring the circumference of the bangle?Yes. You have to mark a point on the bangle, place one end of a thread at the point, then place the rest of the thread firmly round the bangle until you reach the point you started. The length of the thread you use is the length around the bangle. (The cirumference of the circle)We can also mark a point on the bangle and role it along a straight line marking the starting point, until the point on the bangle meets the line again. By that time the bangle has gone one complete round. Then the length of the straight line will be equal to the circumference of the bangle.Alright, do you know how to measure the diameter of the bangle?

- A diameter is a line joining two points on the circumference and passing through the centre.But, if we do not know the centre how do we find the diameter?There are two set squares in the box of instruments. Let us keep the bangle on the ruler and place one of the two sides containing the right angle of each set square on the ruler and the other side touching the bangle as shown in the figure. Then we can read the distance between the two points containing the right angle from the ruler. That is the length of the diameter. Is it true that there is a relationship between the circumference and the diameter of a circle? Let us see whether there is such a relationship.

17.2 The relationship between the circumference and the diameter of a circle

« The length of the boundary around a circle is its circumference (c)'

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The line passing through the centre joining two points on the circumference is its diameter (the length of a diameter - d)

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The distance from the centre to the circumference is the radius (r)«Also, d = 2r «

Collect a few circular shapes that you can find easily. Examples- A bangle, an empty fish tin, yoghurt cups, lids etc.Measure the circumference and the diameter and complete the table given below.

It is seen that a value close to 3.14 is obtained for each item in the or

column.

Accordingly,

This constant value is referred to as p, (writern in English Language as "pi")

A relationship can be formed as =

c pd=A diameter is twice the radius.

That is d 2r =Then, when 2r is substituted for d in the formula c = pd,

c 2pr is obtained=

1 Bangle2 The base of a fish tin3 Lid of a bottle4 Lid of yoghurt cup5 ----------------

Object Circumference (c) Diameter (d) c/d

circumferencediameter

circumferencediameter

= = 3.14 (approximately) (A constant)cd

cd

cd

cd

d ´

r rd

c 2pr=

Activity 1

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p

\

Although is used for p , it is easy to use the value for p in some calculations.

Also is approximately equal to 3.14.

The radius of a circle is 10 cm. Find its circumference.^Take p = 3.14&

The radius (r) of the circle = 10 cmThe circumference (c) of the circle= 2pr

= 2 ́ 3.14 ́ 10 cm= 62.8 cm

The radius of a circle is 3.5 cm, Find its circumference. ̂ Take p = &The radius (r) of the circle = 3.5 cmThe circumference (c) of the circle= 2pr

= 2 ́ ́ 3.5 cm

= 22 cm

If the circumference of a circle is 44 cm, find its radius ̂ Take p = &

The circumference (c) of the circle= 44 cmc = 2pr 2pr = 44cm

= 44cm

= 44cm

=

r = 7cm

3.14 227

227

227

Example 1

Example 2

227

Example 3

227

227

2 ´ ´ r

447

447

744

744

´ r

´ ´ r ´ 44cm

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2

The ratio of the radii of two circles is 2:3. Find the ratio of their circumferences.The ratio of the two radii = 2 : 3

\If the radius of the first circle is 2x, the radius of the second circle is 3x.

Then the circumference of the first circle = 2pr

The circumference of the second circle = 2pr

The ratio of the two circumferences

1. Calculate the circumference of each circle using the information given below'

^p = 3.14&

(i) The radius is 20 cm (ii) The diameter is12 cm(iii)The radius is 9 cm (iv) The diameter is15.2 m

2. Calculate the circumference of each circle using the information given below.

^Take p = &

(i) The radius is 28 cm (ii) The radius is 3.5 cm(iii)The diameter is 10.5 cm (iv) The diameter is 19.6 cm

3. Calculate the radius of the circles with the circumference are given below.(i) The circumference is 88 cm (ii) The circumference is 220 cm(iii)The cirumference is 66 cm (iv) The circumference is 330 m

4. The diameter of a wheel of a motor vehicle is 63 cm. Find the number of complete rotations the wheel does during a journey of 2 km.

Exercise 17.1

The ratio of the radii of two circles and the ratio of their circumferences are equal.

227

Example 4

= 2 ´ ´ (2x)227

= 2 ´ ´ (3x)227

= 2 ´ ´ (2x) :227

2 ´ ´ (3x) 227

= 2 : 3

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5. The length of the minute hand of a clock is 4.2 cm. What is the distance, the tip of the minute hand covers during each of the periods of time given below.

(i) 60 Minutes (ii) 120 Minutes (iii)30 Minutes (iv) 15 Minutes (v) 45 Minutes

6. A semi- circular stage is shown in the given diagram. Its diameter is 7 m. A steel belt is fixed along the circular part. Find the length of the belt.

7. A rough plan of a running track is shown in the diagram. The track consists of a rectangular part and two semi-circular parts.The length of the rectangular part is 112m and the breadth is 56 m. Find the perimeter of the track.

8. The given diagram shows a plan of a grill made of metal bars fitted on to a window. Find the length of the metal bars used for this.

9. Find the perimeter of the following shapes shown in the diagrams. All circular parts in the diagrams are semi-circles.

12

6

9 34.2cm

7m

28 cm28

cm

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10. The diagram shows a part of a machine. This part consists of two circular wheels of two different radii ; touching each other. When one wheel rotates the other wheel too rotates. How many rounds will the smaller wheel rotate, when the larger wheel rotates one round?

11. A plan of a running track is shown here. This consists of a rectangular part and two semicircular parts. The length of the rectangular part is 112 m and the radius of the semi - circular part is 27.5 m. The width of each lane is 1m. The runners are expected to use the midline of the lane when running.

(i) What is the radius of the semi - circular part which the runner in the first lane uses for running?

(ii) Find the length which the runner in the first lane covers when he completes a round.

th(iii)At what distance ahead should the competitor in the 8 lane start the race; st if he is to cover the same distance as that of the competiter in the 1 lane?

12. Assume that you are assigned to prepare a 400m running track for the inter-house sports meet. If the track should have a rectangular part and two semi-circular parts as shown in the diagram, find three sets that can be used for x and y

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(i) (ii)

y

x

14cm

14cm

14cm 7cm

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14 cm"