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Circles, arcs & sectors
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48 questions at your level
Difficulty
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Level 2
Level 3
Level 4
Level 5
Arc length & sector area
20 questions
Lesson
Not started
Mark as done
A sector of a circle has radius
14
14
14
cm and arc length
18.5
18.5
18.5
cm.
Work out the angle of the sector. Give your answer correct to 1 decimal place.
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Level 4
3 marks
Start
→
Mark as done
A sector of a circle has area
45
45
45
cm² and angle
100
°
100°
100°
.
Work out the radius of the circle. Give your answer correct to 3 significant figures.
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Level 4
3 marks
Start
→
Mark as done
A sector of a circle of radius
12
12
12
cm has area
24
π
24\pi
24
π
cm².
(a)
Work out the angle of the sector.
(b)
Work out the arc length of the sector. Give your answer in terms of
π
\pi
π
.
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Level 4
4 marks
Start
→
Mark as done
O
P
Q
OPQ
O
P
Q
is a sector of a circle, centre
O
O
O
, radius
8
8
8
cm.
O
Q
P
8 cm
The length of arc
P
Q
PQ
P
Q
is
6
π
6\pi
6
π
cm.
Work out, in terms of
π
\pi
π
, the area of the sector.
Give your answer in its simplest form.
(4 marks)
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Level 4
4 marks
Start
→
Mark as done
O
P
Q
OPQ
O
P
Q
is a triangle.
O
Q
R
OQR
O
QR
is a sector of a circle, centre
O
O
O
.
O
Q
R
58°
7.4 cm
5.8 cm
118°
P
Calculate the area of the sector
O
Q
R
OQR
O
QR
.
Give your answer correct to 3 significant figures.
(4 marks)
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●
Level 4
4 marks
Start
→
Mark as done
The four vertices of rectangle
P
Q
R
S
PQRS
P
QR
S
lie on a circle, centre
O
O
O
.
11 cm
6.5 cm
P
Q
R
S
O
P
Q
=
11
PQ = 11
P
Q
=
11
cm
Q
R
=
6.5
\qquad QR = 6.5
QR
=
6.5
cm
Work out the circumference of the circle.
Give your answer correct to 3 significant figures.
(4 marks)
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Level 4
4 marks
Start
→
Circle theorems
18 questions
Lesson
Not started
Mark as done
A
A
A
,
B
B
B
,
C
C
C
and
D
D
D
are points on a circle with centre
O
O
O
.
Angle
A
O
C
=
130
°
AOC = 130°
A
O
C
=
130°
, where
A
O
C
AOC
A
O
C
is the angle at the centre standing on the arc
A
B
C
ABC
A
B
C
.
A
B
C
D
ABCD
A
B
C
D
is a cyclic quadrilateral.
(a)
Work out the size of angle
A
D
C
ADC
A
D
C
. Give a reason for your answer.
(b)
Work out the size of angle
A
B
C
ABC
A
B
C
. Give a reason for your answer.
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●
Level 4
6 marks
Start
→
Mark as done
A
A
A
,
B
B
B
and
C
C
C
are points on a circle with centre
O
O
O
.
The
reflex
angle
A
O
B
=
216
°
AOB = 216°
A
O
B
=
216°
.
216°
x
A
B
C
O
Diagram NOT accurately drawn
Work out the size of the angle marked
x
x
x
. Give a reason for your answer.
●●●●
●
Level 4
3 marks
Start
→
Mark as done
A
A
A
,
B
B
B
,
C
C
C
and
D
D
D
are points on a circle with centre
O
O
O
.
A
B
C
D
ABCD
A
B
C
D
is a cyclic quadrilateral.
Angle
A
O
C
=
126
°
AOC = 126°
A
O
C
=
126°
, where angle
A
O
C
AOC
A
O
C
is the angle on the side of the quadrilateral nearest to
B
B
B
.
126°
x
y
A
B
C
D
O
Diagram NOT accurately drawn
(a)
Work out the size of angle
A
D
C
ADC
A
D
C
, marked
x
x
x
. Give a reason for your answer.
(b)
Work out the size of angle
A
B
C
ABC
A
B
C
, marked
y
y
y
. Give a reason for your answer.
●●●●
●
Level 4
4 marks
Start
→
Mark as done
A
A
A
,
B
B
B
,
C
C
C
and
D
D
D
are points on a circle.
A
C
AC
A
C
is a diameter of the circle.
Angle
B
A
C
=
27
°
BAC = 27°
B
A
C
=
27°
.
27°
A
B
C
D
Diagram NOT accurately drawn
Work out the size of angle
A
D
B
ADB
A
D
B
.
Give reasons for your answer. (4 marks)
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●
Level 4
4 marks
Start
→
Mark as done
A
A
A
,
B
B
B
and
C
C
C
are points on a circle with centre
O
O
O
.
T
A
TA
T
A
and
T
B
TB
T
B
are tangents to the circle.
Angle
A
T
B
=
48
°
ATB = 48°
A
T
B
=
48°
.
48°
O
A
B
C
T
Diagram NOT accurately drawn
(a)
Work out the size of angle
A
O
B
AOB
A
O
B
. Give reasons for your answer. (2 marks)
(b)
Work out the size of angle
A
C
B
ACB
A
C
B
. Give a reason for your answer. (2 marks)
●●●●
●
Level 4
4 marks
Start
→
Mark as done
A
A
A
,
B
B
B
,
C
C
C
and
D
D
D
are points on a circle, centre
O
O
O
.
124°
O
A
B
C
D
Angle
A
O
C
=
124
°
AOC = 124°
A
O
C
=
124°
Work out the size of angle
A
B
C
ABC
A
B
C
.
Write down any circle theorems you use.
(4 marks)
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●
Level 4
4 marks
Start
→
Mark as done
The four vertices of rectangle
P
Q
R
S
PQRS
P
QR
S
lie on a circle, centre
O
O
O
.
11 cm
6.5 cm
P
Q
R
S
O
P
Q
=
11
PQ = 11
P
Q
=
11
cm
Q
R
=
6.5
\qquad QR = 6.5
QR
=
6.5
cm
Work out the circumference of the circle.
Give your answer correct to 3 significant figures.
(4 marks)
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●
Level 4
4 marks
Start
→
Tangents, chords & the alternate segment
14 questions
Lesson
Not started
Mark as done
A
A
A
,
B
B
B
and
C
C
C
are points on a circle.
T
A
TA
T
A
is a tangent to the circle at
A
A
A
.
The angle between the tangent
T
A
TA
T
A
and the chord
A
B
AB
A
B
is
63
°
63°
63°
.
Angle
A
B
C
=
71
°
ABC = 71°
A
B
C
=
71°
.
(a)
Write down the size of angle
A
C
B
ACB
A
C
B
. Give a reason for your answer.
(b)
Work out the size of angle
B
A
C
BAC
B
A
C
.
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●
Level 4
4 marks
Start
→
Mark as done
A circle has centre
O
O
O
and radius
25
25
25
cm.
A
B
AB
A
B
is a chord of length
30
30
30
cm.
(a)
Work out the distance from
O
O
O
to the chord
A
B
AB
A
B
.
(b)
M
M
M
is the midpoint of
A
B
AB
A
B
, and the line
O
M
OM
O
M
is extended to meet the circle at
N
N
N
. Work out the length of
M
N
MN
M
N
.
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●
Level 4
5 marks
Start
→
Mark as done
P
T
PT
P
T
is a tangent to a circle with centre
O
O
O
and radius
7
7
7
cm, touching the circle at
T
T
T
.
P
T
=
24
PT = 24
P
T
=
24
cm.
7 cm
24 cm
O
T
P
Diagram NOT accurately drawn
(a)
Work out the length
O
P
OP
O
P
.
(b)
Work out the shortest distance from
P
P
P
to the circle.
●●●●
●
Level 4
5 marks
Start
→
Mark as done
P
A
PA
P
A
and
P
B
PB
P
B
are tangents to a circle with centre
O
O
O
.
Angle
A
O
B
=
138
°
AOB = 138°
A
O
B
=
138°
.
x
138°
O
A
B
P
Diagram NOT accurately drawn
Work out the size of the angle marked
x
x
x
. Give reasons for your answer.
●●●●
●
Level 4
3 marks
Start
→
Mark as done
A
A
A
,
B
B
B
and
C
C
C
are points on a circle with centre
O
O
O
.
T
A
TA
T
A
and
T
B
TB
T
B
are tangents to the circle.
Angle
A
T
B
=
48
°
ATB = 48°
A
T
B
=
48°
.
48°
O
A
B
C
T
Diagram NOT accurately drawn
(a)
Work out the size of angle
A
O
B
AOB
A
O
B
. Give reasons for your answer. (2 marks)
(b)
Work out the size of angle
A
C
B
ACB
A
C
B
. Give a reason for your answer. (2 marks)
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●
Level 4
4 marks
Start
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