Circles (Area)
Circumference Revision
In Book N4-1, Chapter 9, you discovered
that you could calculate the circumference
of a circle using this formula.
C = ! x D
Circumference = 3·14 x Diameter
C = !d
Exercise 1
1.
In this exercise, give your answers to 2 decimal places where necessary.
Calculate the circumference of the clock face and the cake :–
a
b
radius = 17·5 cm
diameter = 30 cm
2.
The diameter of this letter “D” is 8 centimetres.
8 cm
Find the length of the curved part of the D.
3.
The end of this strip of edging is in the shape of a quarter-circle PQR.
P
Q
4.
2·4 cm
2·4 cm
R
a
Calculate the circumference of a circle with radius 2·4 cm.
b
Now find the length of the curved part PR.
c
Finally, calculate the total distance round the edge of the end face.
The rear wheel of this vehicle has a diameter of 120 cm.
a
How far would the vehicle travel in one complete
turn of the wheel ?
(Give your answer in metres).
b
How far would it travel in 200 turns (in metres) ?
c
How many full turns would a wheel need to make if the
vehicle travelled a distance of 1884 metres ?
National 4 Book N4-2
this is page 115
120 cm
Chapter 8 - Circles (Area)
5.
Calculate the total distance around the outsides of these shapes :–
a
b
25 cm
25 cm
11 mm
25 cm
25 cm
17 mm
35 cm
6.
The diagram shows a picture of a large
factory gate.
1·5 m
Calculate the total
perimeter of the gate.
6m
7.
“Scotties” oatcakes come in 2 different shapes.
3·5 cm
7 cm
8·5 cm
Which of the two oatcakes has the larger perimeter ?
(Justify your answer with working).
8.
40 cm
A wooden sign, advertising “Hubert Estate Agents”
consists of a 12 circle on top of a trapezium, as shown.
55 cm
Calculate the perimeter of the sign.
Hubert
Estate
Agents
55 cm
70 cm
9.
This diagram shows 2 paths through a forest.
There is a straight road A —> B —> C and a
path consisting of 2 semicircles.
60 m
A
60 m
National 4 Book N4-2
B
C
a
Calculate the length of the curved route
from A round to B, then round to C.
b
How much longer is this route than the
straight route ?
this is page 116
Chapter 8 - Circles (Area)
Find the Diameter
You can use the formula C = !D in reverse
to find the diameter of a circle when you
already know its circumference. :–
diameter
Example 1 :– Find the diameter of the circle.
C = !D
628 = 3·14 x D
d = 200 cm (as
Can you see that the formula needed
to calculate the diameter is :–
628
3⋅14
= 200)
Circumference
D= C
!
Diameter
Example 2 :– Find the diameter of the
circle with a circumference
of 15·7 centimetres.
628 cm
3·14
C
D= !
D=?
D=
15·7
3·14
D = 5 cm
circum = 15·7 cm
Exercise 2
1.
Find the diameter (D) when the circumference (C) is :–
a
2.
314 cm
b
9·42 m
c
1884 mm.
Find the diameter of each of these rounds of cheese, (to 1 decimal place).
a
b
circumference = 45 cm
c
circumference = 82 cm
circumference = 1·25 m
National 4 Book N4-2
this is page 117
Chapter 8 - Circles (Area)
3.
Calculate, showing all the steps, the diameters of these circles :–
a
b
c
C = 34·54 cm
C = 39·25 cm
C = 3768 cm
4.
The circumference of this “happy face” is 20·41 cm.
Find its diameter.
5.
The circumference of this hole, drilled
through the piece of wood, is 78·5 mm.
Calculate the diameter of the hole.
6.
C = 78·5
mm
This silver plate has a circumference of 132 cm.
Find its diameter.
Remember the link between the diameter and a radius ?
radius =
7.
1
2
diameter
of diameter
radius
Calculate the diameter of each of these circles and then write down the length of
each radius.
a
b
c
C = 80 m
C = 120 m
8.
C = 200 m
This C.D. has a circumference of 40 centimetres.
Find its radius.
9.
A lemon is sliced through its middle.
The circumference of the lemon is 17 centimetres.
Calculate its diameter.
National 4 Book N4-2
this is page 118
Chapter 8 - Circles (Area)
Finding r2
“
”
Finding r2 (Finding “r squared”)
r
* Squaring a number means multiplying the number by itself.
72 = 7 x 7 = 49
Examples :–
42 = 4 x 4 = 16
Example :-
If the radius of a circle is 8 cm,
2
find the value of the “radius squared” (r 2).
r = 8 (cm) = r 2 = 8 x 8 = 64 (cm2)
Exercise 3
1.
Find the value of r 2 given that :–
a
r=3
b
r=2
c
r = 10
d
r=9
e
r=5
f
r = 12
g
r = 25
h
r = 17
i
r = 40
j
r = 120.
Area of a Circle
The Formula for the area of a circle is :Area
A = ! x r2
Area
3·14
radius
r x r
where A is the area, ! is Pi (= 3·14), and r is the radius.
Example :–
Find the area of this circle with a radius 20 metres.
A = ! 2r
=>
=>
Exercise 4
1.
20 m
A = 3·14 x 20 x 20
2
A = 1256 m (square metres)
Give your answers to 2 decimal places where necessary.
Calculate the area for each of these circles :–
a
b
9 cm
4 cm
National 4 Book N4-2
this is page 119
Chapter 8 - Circles (Area)
2.
Calculate the area of this circle.
3.
For each of the following circles, set down the three
lines of working and calculate their areas :–
a
7 cm
b
c
3 cm
25 cm
13 cm
d
e
f
0·6 cm
4·5 cm
4.
10·35 cm
Careful !!
In this question, you are given the diameter (not the radius).
Copy and complete to calculate the area :–
Step 1 :–
radius
=
=
1
2
1
2
of diameter
of 30 cm
= .... cm
Step 2 :–
A =!xr
area
30 cm
2
=> A = 3·14 x r x r
=> A = .... cm2
5.
For this circle :–
18 cm
6.
Calculate the area for
each of these circles :–
a
Halve the diameter to get the radius.
b
Use this to calculate its area (set down !)
a
b
2·4 m
National 4 Book N4-2
this is page 120
1·04 m
Chapter 8 - Circles (Area)
7.
For each of the following circles, write down the radius, then set down the three lines
of working needed to calculate their areas :–
a
b
c
4 cm
50 cm
14 cm
d
e
7 cm
8.
f
9 mm
40 cm
The diameter of an old Vinyl L. P. record is 30 centimetres.
Calculate its area.
9.
30 cm
The top of a tin of beans has a diameter of 9·2 cm.
Calculate the area of the top of the tin.
10. This design has a diameter of 36 mm.
36 mm
What is its area ?
11.
The radius of both of these metal symbols is 17·5 cm.
Calculate the area of metal needed to make both symbols.
12. This no smoking sign has a diameter of 56 mm.
56 mm
Calculate its area.
13.
The glass see-through door on the washing machine
has a radius of 19 centimetres.
Calculate the area of the glass door.
National 4 Book N4-2
this is page 121
Chapter 8 - Circles (Area)
Area of Semi-circles, part circles ...
A Semi-circle is simply a half-circle.
To find its area, simply find the area of the whole circle – then halve your answer.
A Quarter-circle
To find its area, simply find the area of the whole circle – then divide your answer by 4.
Exercise 5
Give your answers in this exercise to 2 decimal places where necessary.
1.
Shown is a circle with radius 12 cm.
12 cm
2.
a
Calculate the area of the whole circle.
b
Now halve your answer to obtain the
shaded area, (the semi-circle).
Shown is a semi-circular garden pond
where goldfish are being bred.
The semi-circular pond has radius 4 metres.
Calculate the area of the pond.
4m
3.
Shown below is a doorway, consisting of a rectangle with a semi-circle on top.
3m
2m
4.
a
Calculate the area of the rectangle.
b
Write down the diameter of the semi-circle and its radius.
c
Calculate the area of the whole circle and then the area of the semi-circle.
d
Now calculate the area of the whole doorway.
Calculate the area of this shape.
Decide how many parts you want
to break it into first.
30 m
30 m
(Hint :– A rectangle and a circle).
70 m
National 4 Book N4-2
this is page 122
Chapter 8 - Circles (Area)
5.
Here is a quarter-circle with radius 7 cm.
a
Find the area of the whole circle which has a radius 7 cm.
b
Now (÷ 4), to find the area of the quarter-circle.
7 cm
6.
7.
Calculate the yellow area shown here.
22 m
This shape consists of a right angled triangle and a quarter circle.
a
Calculate the area of the triangle.
b
Calculate the area of the quarter circle.
c
Calculate the area of the whole shape.
16 cm
30 cm
16 cm
10 cm
8.
15 cm
A semi-circular hole is cut from the
edge of this rectangular piece of
wood to fit around a circular pipe.
30 cm
9.
a
Calculate the area of the rectangular piece of wood.
b
Calculate the area of the semi-circle.
c
Calculate the area of the remaining piece of wood.
This metal washer consists of a circular piece of steel
with a circular hole in its centre.
a
Calculate the area of the large circle, radius 18 mm.
b
Calculate the area of the smaller circle.
c
Calculate the grey area representing the
area of one face of the washer.
7 mm
18 mm
10. 4 holes are drilled out of a thin piece of wood.
6 cm
6 cm
6 cm
6 cm
10 cm
35 cm
a
Calculate the area of the original rectangular piece of wood.
b
Calculate the area of one of the holes and then write the area of all 4 holes.
c
Calculate the area of the remaining wood.
National 4 Book N4-2
this is page 123
Chapter 8 - Circles (Area)
Circles – Harder Questions
Exercise 6
1.
When seated at a table, a person needs approximately
70 cm of “table-edge” to be comfortable.
70 cm
This large circular table has a diameter of 340 cm.
340 cm
Can 14 people be seated comfortably round this table ?
(Justify your answer with working).
2.
The edging for a garden path is made of metal wire, shaped to form
semi-circular loops as shown.
Each loop is 24 centimetres across.
24 cm
3.
a
Find the number of loops needed for one side of the path which is 12 metres long.
b
What length of wire is needed to make all of these loops for one side of this path ?
This “heart - shape” consists of three semi-circle.
The diameter of the large one
is 24 centimetres and the two
smaller ones are identical.
24 cm
a
Calculate the area of the shape.
b
Calculate the perimeter of the shape.
4.
A company logo uses a square with 4 quarter
circles as shown.
12 cm
National 4 Book N4-2
The square has a side length of 12 cm.
a
Calculate the area of the logo.
b
Calculate the perimeter of the logo.
this is page 124
Chapter 8 - Circles (Area)
5.
A church window is shown. It has a wooden frame.
It consists of a semi-circle on top of a rectangle.
1m
Calculate :–
6.
a
the length of the semi-circular
wooden frame at the top.
b
the amount of wood surrounding
the 3 sides of the rectangle.
c
the total amount of wood used
for the window frame.
d
the area of the rectangular glass window.
e
the area of the semi-circular window.
f
the total area of glass used.
3m
2m
A sports field is rectangular with semi-circular ends.
100 m
30 m
Calculate the circumference of the circle which is made by
putting together the two semi-circular ends.
b
Calculate the perimeter of the sports field.
The top face of an eraser is shown.
It is in the shape of a rectangle with two
semi-circular ends.
a
Write down the radius of each of the end parts.
b
Calculate the total area of the end parts.
c
Find the area of the whole face of the eraser.
National 4 Book N4-2
this is page 125
ERABIM
ERASERS
7.
a
4 cm
2 cm
Chapter 8 - Circles (Area)
8.
9.
A new wafer-thin mint is designed as shown. It consists of a square with a hole in
the centre. The square has sides of 3 cm in length and the hole is 1 cm in diameter.
a
the area of the square.
b
the area of the hole.
c
the area of the blue minty part. (shaded)
3 cm
A stylish bathroom floor mat is shown. It is rectangular in shape with a semi-circle
cut out to allow the mat to fit around a sink unit.
Calculate the area of :–
50 cm
100 cm
a
the mat before the cut out.
b
the semi-circle.
c
the actual finished floor mat.
200 cm
10. This lawn is rectangular, but has a
quarter-circle cut out at one corner
to be used for bedding plants.
•
Write down the length represented
in the diagram by :–
(i)
A
3m
C
A fence is to built around the lawn part.
a
B
A
4m
14 m
(ii) B.
b
Calculate the length of the quarter-circle C.
c
What is the total length of fencing required ?
d
Fencing costs £11·50 per metre. What will be the total cost for this fence ?
11. The side view of a new car hoover is shown.
The side, made of strong plastic, is rectangular with a quarter-circle nozzle.
Calculate the area of :–
a
the rectangular side.
b
the quarter circle nozzle.
c
the whole side of the hoover.
CARGLO
20 cm
NOZZLE
40 cm
National 4 Book N4-2
this is page 126
Chapter 8 - Circles (Area)
What Have I Learned ?
1.
Calculate the circumference of each circle :–
a
b
20 cm
2.
13 cm
The circumference of this circle is 942 cm.
a
Calculate its diameter.
b
Now write down its radius.
3.
circumference = 942 cm
Calculate the length of this semi-circle,
which has a diameter of 60 mm.
60 mm
4.
Calculate the area of these 2 circles :–
a
15 m
b
4 cm
5.
The diameter of this circle is 90 mm.
a
Write down its radius.
b
Calculate its area.
6.
90 mm
Calculate the area of this semi-circle.
9 mm
7.
Calculate the area of this quarter-circle.
8.
By breaking this shape
down into various “bits”,
calculate the area of
each “bit” and find the
area of the whole shape.
National 4 Book N4-2
27 cm
20 cm
12 cm
12 cm
this is page 127
15 cm
Chapter 8 - Circles (Area)