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Naming the parts of a circle

circumference: distance around the outside of a circle

radius: distance from the centre of the circle to the circumference

diameter: distance across the width of the circle through the centre

A circle is a set of points equidistant from its centre.

radius

circumference

diameter

centre

chord

tangent

chord: line segment with two endpoints on the circle

tangent: straight line that touches the circle at a single point


Arcs and sectors

An arc is a part ofthe circumference.

When an arc is bounded by two radii a sector is formed.

arc

A minor arc is shorter thana semicircle and a major arc is longer than a semicircle.

sector

segmentA segment is a region bounded by a chord and an arc lying between the chord's endpoints.


Arcs and sectors


π = 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196 (to 200 decimal places)!

The value of π

For any circle, the circumference is always just over three times bigger than the diameter.

The exact number is called π (pi).

The symbol π is used because the number cannot be written exactly.


Approximations for the value of π

When performing calculations involving the value π, an approximation for the value is required.

For a rough approximation use 3.

Better approximations are 3.14 or .227

The π button on a calculator can also be used.

Most questions will indicate which approximation to use.

When a calculation has lots of steps, write π as a symbol throughout and evaluate it at the end, if necessary.


The circumference of a circle

Circumference of a circle = πD

9.5 cm C = 3.14 × 9.5

= 29.83 cm

Use π = 3.14 to find the circumference of this circle.


The circumference of a circle


?

Finding the radius

12 cm

r =C

2π

12

2 × 3.14=

= 1.91 cm (to 2 d.p.)

Use π = 3.14 to find the radius of this circle.

How can this be arrangedto make r the subjectof the formula?

C = 2πr


This running track is made up of a square and 2 semicircles.

x

There is to be a race of one lap around the track.

For the race to be fair, how far must the runner on the outside lane start in front of the runner on the inside lane?

Running track

It has 6 lanes that are each 1 m wide and the straight sections of the 6 lanes are 100 m long.

The total width of the track (x) is 100 m.


Runner in the middleof the outside lane.

x

Runner in the middleof the inside lane.

Both runners run the same distance on the 2 straights.

Running track

Radius of semi-circle = 49.5 m Radius of semi-circle = 44.5 m

C = 2πrC = 2 × 3.14 × 49.5C = 310.86 m

C = 2πrC = 2 × 3.14 × 44.5C = 279.46 m

The runner in the outside lane would need tostart 310.86 – 279.46 = 31.4 m in front of therunner in the inside lane for the race to be fair.

Running track


Formula for the area of a circle

The area of a circle can be found using the formula:

rArea of a circle = π × r × r

Area of a circle = πr2

7 cmA = πr2

= 3.14 × 72

= 153.86 cm2

Use π = 3.14 to find the area of this circle.


The area of a circle


The pond designer The pond designer


Sort the formulae


Finding the length of an arc

An arc is a section of the circumference.

The length of arc AB is a fraction of the length of the circumference.

To work out what fraction of the circumference it is, look at the angle at the centre.

In this example, there is a 90° angle at the centre.

What is the length of arc AB?

A

B

6 cm



A

B

6 cm

The arc length is of the circumference of the circle.

14

90°360°

=14

Length of arc AB = × 2πr14

= 14 × 2π × 6

Length of arc AB = 9.42 cm (to 2 d.p.)

What is the length of arc AB?

For the arc length:



For any circle with radius r and angle at the centre θ...

A B

r θ

arc length AB =θ

360× 2πr

Arc length AB =2πrθ360

=πrθ180

This is the circumferenceof the circle.


The perimeter of shapes made from arcs

The perimeter of this shape is made from three semi-circles.

Perimeter = 12 × π × 6 +12 × π × 4 +12 × π × 2

= 6π cm= 18.85 cm (to 2 d.p.)

Perimeter = 19 × π × 12 +19 × π × 6 +3 + 3

= 2π + 6= 12.28 cm (to 2 d.p.)

40°360°

=19

40°

Find the perimeter of these shapes on a cm square grid:


What would the length of the arc be now?

If the arm is 50 cm long,what is the length of the arc?

Daleks

This is an aerial view of a Dalek’s laser arm. It can move through 40°.

The range of the laser beam is 25 times the length of the arm itself.

If Dr Who was standing 13 maway, would he be in range?

Daleks


Finding the area of a sector

72°6 cm

Area of the sector =72°

360°× π × 62

= × π × 621

5

= 22.62 cm2 (to 2 d.p.)

This method can be used to find the area of any sector.

What is the area of this sector?



For any circle with radius r and angle at the centre θ,

Area of sector AOB =θ

360× πr2

A B

r θ

O

This is the areaof the circle.

Area of sector AOB =πr2θ360


40°

The area of shapes made from sectors

Area = 12 × π × 32

= 3π cm2

= 9.42 cm2 (to 2 d.p.)

Area = 19 × π × 62

= 6.98 cm2 (to 2 d.p.)

19

40°360°

=

= cm219 × π × 20

12 × π × 12+12 × π × 22– 1

9 × π × 42–

Find the area of these shapes on a cm square grid:

40°


The Pizza Shop

The Pizza Shop wants to make a new pizza called the“Eight Taste Pizza” where there is one slice of each topping.

45360

× 3.14 × 3 × 3 =

360°÷ 8 = 45°

Area of Small pizza sector =

= 4 square inches to nearest square inch

Now try the medium and large.

36045 × πr2

3.53 square inches

There are 3 sizes of pizza: small, mediumand large (6, 9 and 12 inch diameters).Calculate the area of the sectors sothe amount of topping requiredcan be determined.

The Pizza Shop

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