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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
The diagram shows a sector of a circle of radius
6
6
6
cm. The angle of the sector is
120
°
120°
120°
.
O
6 cm
120°
(a)
Work out the arc length of the sector. Give your answer correct to 1 decimal place.
(b)
Work out the area of the sector. Give your answer correct to 1 decimal place.
●●
●●●
Level 2
4 marks
Start
→
Mark as done
A semicircle has radius
5
5
5
cm. Work out the length of its curved edge. Give your answer to 1 decimal place.
●●
●●●
Level 2
2 marks
Start
→
Mark as done
A circle has radius
7.5
7.5
7.5
cm.
(a)
Work out the area of the circle. Give your answer correct to 1 decimal place. [2]
A semicircle is cut from the circle.
(b)
Work out the perimeter of the semicircle. Give your answer correct to 1 decimal place. [2]
●●
●●●
Level 2
4 marks
Start
→
Mark as done
A sector of a circle has radius
9
9
9
cm and angle
40
°
40°
40°
.
(a)
Work out the area of the sector. Give your answer correct to 3 significant figures. [2]
(b)
Work out the perimeter of the sector. Give your answer correct to 3 significant figures. [2]
●●
●●●
Level 2
4 marks
Start
→
Mark as done
The diagram shows a sector of a circle of radius
12
12
12
cm. The angle of the sector is
75
°
75°
75°
.
O
12 cm
75°
Diagram NOT accurately drawn
(a)
Work out the arc length of the sector. Give your answer correct to 1 decimal place.
(b)
Work out the area of the sector. Give your answer correct to 1 decimal place.
●●●
●●
Level 3
4 marks
Start
→
Mark as done
The diagram shows a sector of a circle of radius
8
8
8
cm. The angle of the sector is
135
°
135°
135°
.
O
8 cm
135°
Diagram NOT accurately drawn
Work out the perimeter of the sector. Give your answer correct to 1 decimal place.
●●●
●●
Level 3
3 marks
Start
→
Mark as done
A sector of a circle has radius
9
9
9
cm and angle
40
°
40°
40°
.
(a)
Work out the arc length of the sector. Give your answer in terms of
π
\pi
π
.
(b)
Work out the area of the sector. Give your answer in terms of
π
\pi
π
.
●●●
●●
Level 3
4 marks
Start
→
Mark as done
The diagram shows a sector of a circle of radius
5.2
5.2
5.2
cm. The angle of the sector is
210
°
210°
210°
.
O
5.2 cm
210°
Work out the
perimeter
of the sector. Give your answer correct to 1 decimal place.
●●●
●●
Level 3
3 marks
Start
→
Mark as done
The diagram shows a sector of a circle, centre
O
O
O
, radius
6.5
6.5
6.5
cm. The angle of the sector is
110
°
110°
110°
.
O
6.5 cm
110°
Diagram NOT accurately drawn
(a)
Work out the area of the sector. Give your answer correct to 3 significant figures. (3 marks)
(b)
Work out the arc length of the sector. Give your answer correct to 3 significant figures. (1 mark)
●●●
●●
Level 3
4 marks
Start
→
Mark as done
The diagram shows a sector of a circle, centre
O
O
O
.
The radius of the circle is
8
8
8
cm and the angle of the sector is
140
°
140°
140°
.
O
8 cm
140°
Diagram NOT accurately drawn
Work out the perimeter of the sector.
Give your answer correct to 3 significant figures. (4 marks)
●●●
●●
Level 3
4 marks
Start
→
Mark as done
Here is a plan of a play area.
The play area is a rectangle with a semicircle joined to one side.
15 m
9 m
The whole play area will be covered with bark chippings.
Bark chippings are sold in bags.
Each bag of bark chippings covers an area of
4
4
4
m²
(a)
Work out the number of bags of bark chippings needed.
(4 marks)
It turns out that each bag of bark chippings only covers an area of
3.5
3.5
3.5
m²
(b)
How does this affect your answer to part (a)?
(1 mark)
●●●
●●
Level 3
5 marks
Start
→
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.
●●●●
●
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.
●●●●
●
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
.
●●●●
●
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)
●●●●
●
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)
●●●●
●
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)
●●●●
●
Level 4
4 marks
Start
→
Mark as done
The diagram shows a circle of radius
10
10
10
cm. A chord divides the circle, and the angle at the centre is
80
°
80°
80°
.
O
10 cm
80°
Diagram NOT accurately drawn
Work out the area of the shaded segment. Give your answer correct to 1 decimal place.
●●●●●
Level 5
4 marks
Start
→
Mark as done
A circle has radius
7
7
7
cm. The minor arc
A
B
AB
A
B
has length
6.5
6.5
6.5
cm.
(a)
Work out the angle of the minor sector
O
A
B
OAB
O
A
B
. Give your answer correct to 1 decimal place.
(b)
Work out the area of the
major
sector. Give your answer correct to 3 significant figures.
●●●●●
Level 5
5 marks
Start
→
Mark as done
The diagram shows a circle, centre
O
O
O
, radius
r
r
r
cm, and two regular octagons.
r cm
O
The sides of the larger octagon are tangents to the circle.
The vertices of the smaller octagon lie on the circle.
By considering perimeters, show that
4
2
−
2
<
π
<
8
(
2
−
1
)
4\sqrt{2 - \sqrt{2}} < \pi < 8(\sqrt{2} - 1)
4
2
−
2
<
π
<
8
(
2
−
1
)
(4 marks)
●●●●●
Level 5
4 marks
Start
→
Circle theorems
18 questions
Lesson
Not started
Mark as done
A
A
A
,
B
B
B
and
C
C
C
are points on a circle with centre
O
O
O
.
Angle
A
B
C
=
36
°
ABC = 36°
A
B
C
=
36°
.
36°
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 2
3 marks
Start
→
Mark as done
P
P
P
,
Q
Q
Q
and
R
R
R
are points on a circle with centre
O
O
O
.
P
Q
PQ
P
Q
is a diameter.
Angle
Q
P
R
=
27
°
QPR = 27°
QP
R
=
27°
.
27°
x
P
Q
R
O
Diagram NOT accurately drawn
Work out the size of the angle marked
x
x
x
. You must give reasons for each stage of your working.
●●●
●●
Level 3
4 marks
Start
→
Mark as done
A
B
C
D
ABCD
A
B
C
D
is a cyclic quadrilateral.
Angle
D
A
B
=
104
°
DAB = 104°
D
A
B
=
104°
and angle
A
B
C
=
88
°
ABC = 88°
A
B
C
=
88°
.
104°
88°
x
y
A
B
C
D
Diagram NOT accurately drawn
(a)
Work out the size of the angle marked
x
x
x
. Give a reason for your answer.
(b)
Work out the size of the angle marked
y
y
y
.
●●●
●●
Level 3
4 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
.
Angle
A
C
B
=
41
°
ACB = 41°
A
C
B
=
41°
.
41°
x
A
B
C
D
Diagram NOT accurately drawn
(a)
Write down the size of the angle marked
x
x
x
. Give a reason for your answer.
(b)
Work out the size of angle
A
O
B
AOB
A
O
B
. Give a reason for your answer.
●●●
●●
Level 3
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
.
Angle
O
A
B
=
26
°
OAB = 26°
O
A
B
=
26°
.
26°
x
A
B
C
O
Diagram NOT accurately drawn
(a)
Work out the size of angle
A
O
B
AOB
A
O
B
.
(b)
Work out the size of the angle marked
x
x
x
. Give a reason for your answer.
●●●
●●
Level 3
4 marks
Start
→
Mark as done
A
A
A
,
B
B
B
,
C
C
C
and
D
D
D
are points on a circle.
3x°
(x + 20)°
A
B
C
D
Diagram NOT accurately drawn
Angle
D
A
B
=
3
x
°
DAB = 3x°
D
A
B
=
3
x
°
and angle
B
C
D
=
(
x
+
20
)
°
BCD = (x + 20)°
B
C
D
=
(
x
+
20
)
°
.
Work out the value of
x
x
x
. Give a reason for your answer. (3 marks)
●●●
●●
Level 3
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
. Angle
A
O
B
=
140
°
AOB = 140°
A
O
B
=
140°
.
140°
A
B
C
O
Diagram NOT accurately drawn
(a)
Work out the size of angle
A
C
B
ACB
A
C
B
. Give a reason for your answer. (2 marks)
(b)
Work out the size of angle
O
A
B
OAB
O
A
B
. Give a reason for your answer. (2 marks)
●●●
●●
Level 3
4 marks
Start
→
Mark as done
T
A
TA
T
A
and
T
B
TB
T
B
are tangents to a circle with centre
O
O
O
.
C
C
C
is a point on the circle. Angle
A
T
B
=
64
°
ATB = 64°
A
T
B
=
64°
.
64°
A
B
C
T
O
Diagram NOT accurately drawn
(a)
Work out the size of angle
A
O
B
AOB
A
O
B
. Give reasons for your answer. [2]
(b)
Work out the size of angle
A
C
B
ACB
A
C
B
. Give a reason for your answer. [2]
●●●
●●
Level 3
4 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
.
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.
●●●●
●
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)
●●●●
●
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)
●●●●
●
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)
●●●●
●
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
.
Angle
A
C
B
=
58
°
ACB = 58°
A
C
B
=
58°
.
58°
x
A
B
C
O
Diagram NOT accurately drawn
Work out the size of the angle marked
x
x
x
(angle
O
A
B
OAB
O
A
B
). You must give reasons for each stage of your working.
●●●●●
Level 5
4 marks
Start
→
Mark as done
A
A
A
,
B
B
B
,
C
C
C
and
D
D
D
are points on a circle. The chords
A
C
AC
A
C
and
B
D
BD
B
D
cross at the point
E
E
E
.
Angle
A
D
B
=
40
°
ADB = 40°
A
D
B
=
40°
and angle
D
B
C
=
33
°
DBC = 33°
D
B
C
=
33°
.
40°
x
33°
A
B
C
D
E
Diagram NOT accurately drawn
(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
A
E
B
AEB
A
E
B
, marked
x
x
x
.
●●●●●
Level 5
4 marks
Start
→
Mark as done
P
P
P
,
Q
Q
Q
and
R
R
R
are points on a circle, centre
O
O
O
.
S
P
T
SPT
S
P
T
is the tangent to the circle at
P
P
P
.
67°
49°
O
P
Q
R
T
S
Angle
T
P
Q
=
67
°
TPQ = 67°
T
P
Q
=
67°
Angle
R
P
S
=
49
°
RPS = 49°
R
P
S
=
49°
Work out the size of angle
O
Q
R
OQR
O
QR
.
Write down any circle theorems that you use.
(4 marks)
●●●●●
Level 5
4 marks
Start
→
Tangents, chords & the alternate segment
14 questions
Lesson
Not started
Mark as done
The diagram shows a circle with centre
O
O
O
and radius
13
13
13
cm.
A
B
AB
A
B
is a chord of length
24
24
24
cm.
24 cm
13 cm
A
B
O
Diagram NOT accurately drawn
Work out the distance from the centre
O
O
O
to the chord
A
B
AB
A
B
.
●●●
●●
Level 3
3 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
P
B
=
48
°
APB = 48°
A
P
B
=
48°
.
48°
A
B
P
O
Diagram NOT accurately drawn
(a)
Write down the size of angle
O
A
P
OAP
O
A
P
. Give a reason for your answer.
(b)
Work out the size of angle
A
O
B
AOB
A
O
B
.
●●●
●●
Level 3
4 marks
Start
→
Mark as done
P
A
PA
P
A
is a tangent to a circle with centre
O
O
O
and radius
8
8
8
cm. The point
A
A
A
is where the tangent touches the circle.
O
P
=
17
OP = 17
O
P
=
17
cm.
Work out the length of
P
A
PA
P
A
.
●●●
●●
Level 3
3 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
, touching the circle at
A
A
A
and at
B
B
B
.
P
A
=
9
PA = 9
P
A
=
9
cm.
9 cm
O
A
B
P
Diagram NOT accurately drawn
(a)
Write down the length of
P
B
PB
P
B
. Give a reason for your answer.
(b)
Write down the size of angle
O
A
P
OAP
O
A
P
. Give a reason for your answer.
●●●
●●
Level 3
4 marks
Start
→
Mark as done
T
A
TA
T
A
and
T
B
TB
T
B
are tangents to a circle with centre
O
O
O
.
C
C
C
is a point on the circle. Angle
A
T
B
=
64
°
ATB = 64°
A
T
B
=
64°
.
64°
A
B
C
T
O
Diagram NOT accurately drawn
(a)
Work out the size of angle
A
O
B
AOB
A
O
B
. Give reasons for your answer. [2]
(b)
Work out the size of angle
A
C
B
ACB
A
C
B
. Give a reason for your answer. [2]
●●●
●●
Level 3
4 marks
Start
→
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
.
●●●●
●
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
.
●●●●
●
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)
●●●●
●
Level 4
4 marks
Start
→
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
.
Angle
T
A
B
=
58
°
TAB = 58°
T
A
B
=
58°
and
C
A
=
C
B
CA = CB
C
A
=
C
B
.
58°
x
A
B
C
T
Diagram NOT accurately drawn
(a)
Write down the size of angle
A
C
B
ACB
A
C
B
, marked
x
x
x
. Give a reason for your answer.
(b)
Work out the size of angle
C
A
B
CAB
C
A
B
. Give a reason for your answer.
●●●●●
Level 5
4 marks
Start
→
Mark as done
A circle has centre
O
O
O
. The chord
A
B
AB
A
B
has length
16
16
16
cm, and the perpendicular distance from
O
O
O
to
A
B
AB
A
B
is
6
6
6
cm.
16 cm
6 cm
O
A
B
Diagram NOT accurately drawn
(a)
Work out the radius of the circle.
The point
P
P
P
is
26
26
26
cm from
O
O
O
.
P
T
PT
P
T
is a tangent to the circle, touching it at
T
T
T
.
(b)
Work out the length
P
T
PT
P
T
.
●●●●●
Level 5
5 marks
Start
→
Mark as done
R
R
R
and
S
S
S
are points on a circle, centre
O
O
O
.
The tangent to the circle at
R
R
R
and the tangent to the circle at
S
S
S
meet at the point
T
T
T
.
O
R
S
T
Prove that
R
T
=
S
T
RT = ST
R
T
=
S
T
(4 marks)
●●●●●
Level 5
4 marks
Start
→
Mark as done
P
P
P
,
Q
Q
Q
and
R
R
R
are points on a circle, centre
O
O
O
.
S
P
T
SPT
S
P
T
is the tangent to the circle at
P
P
P
.
67°
49°
O
P
Q
R
T
S
Angle
T
P
Q
=
67
°
TPQ = 67°
T
P
Q
=
67°
Angle
R
P
S
=
49
°
RPS = 49°
R
P
S
=
49°
Work out the size of angle
O
Q
R
OQR
O
QR
.
Write down any circle theorems that you use.
(4 marks)
●●●●●
Level 5
4 marks
Start
→