Circles

Shape and Space

The value of π

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

The exact number is called π (pi).

We use the symbol π because the number cannot be written exactly.

π = 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196 (to 200 decimal places)!

Approximations for the value of π

When we are doing calculations involving the value π we have to use an approximation for the value.

Generally, we use the approximation 3.14

We can also use the π button on a calculator.

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

The circumference of a circle

For any circle,

π =circumference

diameter

or,

We can rearrange this to make a formula to find the circumference of a circle given its diameter.

C = πd

π =Cd

Circle circumference and diameter

The circumference of a circle

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

C = πd8 cm

= 3.14 × 8

= 25.12 cm

Finding the circumference given the radius

The diameter of a circle is two times its radius, or

C = 2πr

d = 2r

We can substitute this into the formula

C = πd

to give us a formula to find the circumference of a circle given its radius.

The circumference of a circle

Use π = 3.14 to find the circumference of the following circles:

C = πd4 cm

= 3.14 × 4

= 12.56 cm

C = 2πr9 m

= 2 × 3.14 × 9

= 56.52 m

C = πd

23 mm = 3.14 × 23

= 72.22 mm

C = 2πr58 cm

= 2 × 3.14 × 58

= 364.24 cm

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Finding the radius given the circumference

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

C = 2πr12 cm

How can we rearrange this to make r the subject of the formula?

r =C2π

122 × 3.14

=

= 1.91 cm (to 2 d.p.)

Find the perimeter of this shape

Use π = 3.14 to find perimeter of this shape.

The perimeter of this shape is made up of the circumference of a circle of diameter 13 cm and two lines of length 6 cm.

6 cm13 cm

Perimeter = 3.14 × 13 + 6 + 6

= 52.82 cm

Circumference problem

The diameter of a bicycle wheel is 50 cm. How many complete rotations does it make over a distance of 1 km?

50 cm

The circumference of the wheel

= 3.14 × 50

Using C = πd and π = 3.14,

= 157 cm

The number of complete rotations

= 100 000 ÷ 157

= 637

1 km = 100 000 cm

Formula for the area of a circle

We can find the area of a circle using the formula

radius

Area of a circle = πr2

Area of a circle = π × r × r

or

Area of a circle

The circumference of a circle

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

A = πr24 cm

= 3.14 × 4 × 4

= 50.24 cm2

Finding the area given the diameter

The radius of a circle is half of its radius, or

We can substitute this into the formula

A = πr2

to give us a formula to find the area of a circle given its diameter.

r = d2

A = πd2

4

The area of a circle

Use π = 3.14 to find the area of the following circles:

A = πr22 cm

= 3.14 × 22

= 12.56 cm2

A = πr2

10 m= 3.14 × 52

= 78.5 m2

A = πr2

23 mm = 3.14 × 232

= 1661.06 mm2

A = πr2

78 cm= 3.14 × 392

= 4775.94 cm2

Find the area of this shape

Use π = 3.14 to find area of this shape.

The area of this shape is made up of the area of a circle of diameter 13 cm and the area of a rectangle of width 6 cm and length 13 cm.

6 cm13 cm Area of circle = 3.14 × 6.52

= 132.665 cm2

Area of rectangle = 6 × 13

= 78 cm2

Total area = 132.665 + 78

= 210.665 cm2