circles shape and space. the value of π for any circle the circumference is always just over three...
TRANSCRIPT
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
?
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 =C
2π
12
2 × 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