Circle Angles, Arcs & Chords — Drills
Circle Angles, Arcs & Chords has 13 drill skills. Each one generates fresh questions every time, with a hint and full worked steps, so you can keep practising a single skill until it settles.
What these drills cover
These questions use 11 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Vertex — A flat surface; a line where two faces meet; a corner point.
- Supplementary — Two angles summing to 90° / to 180°.
- Radius — Centre to edge. Half the diameter.
- Arc — Part of the edge; a "pizza slice" bounded by two radii.
- Congruent — Same size and same shape. Written △ ABC ≅ △ DEF.
- Perpendicular — Slopes multiply to -1 — negative reciprocals.
- Central angle — An angle whose vertex is the centre. Equal to its arc.
- Inscribed angle — An angle whose vertex is on the circle. Half its arc.
- Chord — A segment joining two points on the circle.
- Tangent — A line touching the circle at exactly one point.
- Secant — A line that cuts the circle at two points.
Central
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A central angle cuts an arc of 149°. Find the angle.
Worked answer
149
Vertex at the centre, so angle = arc.
149°
Problem 2
A central angle cuts an arc of 34°. Find the angle.
Worked answer
34
Vertex at the centre, so angle = arc.
34°
Inscribed
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
An inscribed angle cuts an arc of 46°. Find the angle.
Worked answer
23
46 ÷ 2 = 23°
Problem 2
An inscribed angle cuts an arc of 82°. Find the angle.
Worked answer
41
82 ÷ 2 = 41°
Semicircle
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
An angle is inscribed in a semicircle. Find its measure.
Worked answer
90
The arc is 180°.
Half is 90° — Thales' theorem.
Vertex
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
An angle with its vertex at the centre equals
- its arc
- half the difference of the arcs
- half its arc
- half the sum of the two arcs
Worked answer
its arc
Vertex at the centre.
So the angle is its arc.
Problem 2
An angle with its vertex on the circle equals
- half the difference of the arcs
- half the sum of the two arcs
- half its arc
- its arc
Worked answer
half its arc
Vertex on the circle.
So the angle is half its arc.
Outside
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Two secants from an external point cut arcs of 170° (far) and 90° (near). Find the angle.
Worked answer
40
170 - 90 = 80
80 ÷ 2 = 40°
Problem 2
Two secants from an external point cut arcs of 90° (far) and 26° (near). Find the angle.
Worked answer
32
90 - 26 = 64
64 ÷ 2 = 32°
Inside
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Two chords cross inside a circle cutting arcs of 40° and 84°. Find the angle.
Worked answer
62
40 + 84 = 124
124 ÷ 2 = 62°
Problem 2
Two chords cross inside a circle cutting arcs of 64° and 68°. Find the angle.
Worked answer
66
64 + 68 = 132
132 ÷ 2 = 66°
Tangent
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A tangent touches a circle of radius 16 at T. From external point P, OP = 34. Find PT.
Worked answer
30
△ OTP has a right angle at T.
PT2 = 342 - 162 = 900
PT = 30
Problem 2
A tangent touches a circle of radius 5 at T. From external point P, OP = 13. Find PT.
Worked answer
12
△ OTP has a right angle at T.
PT2 = 132 - 52 = 144
PT = 12
Chords
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Two chords cross so the pieces are 3, 8 and 6, x. Find x.
Worked answer
4
3 × 8 = 6x
24 = 6x
x = 4
Problem 2
Two chords cross so the pieces are 7, 5 and 7, x. Find x.
Worked answer
5
7 × 5 = 7x
35 = 7x
x = 5
Same Arc
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Two inscribed angles stand on the same arc. One is 71°. Find the other.
Worked answer
71
Angles on the same arc are congruent.
71°
Problem 2
Two inscribed angles stand on the same arc. One is 21°. Find the other.
Worked answer
21
Angles on the same arc are congruent.
21°
Tangentradius
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A tangent meets the radius at the point of contact at
- 180°
- 90°
- 45°
- any angle
Worked answer
90°
A tangent is perpendicular to the radius at the point of contact.
That is why tangent problems turn into right triangles.
Cyclic
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
In a cyclic quadrilateral one angle is 62°. Find the opposite angle.
Worked answer
118
180 - 62 = 118°
Problem 2
In a cyclic quadrilateral one angle is 88°. Find the opposite angle.
Worked answer
92
180 - 88 = 92°
Solving
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
An inscribed angle is (4x)° and its arc is (18x - 130)°. Find x.
Worked answer
13
18x - 130 = 2(4x)
10x = 130
x = 13
Problem 2
An inscribed angle is (4x)° and its arc is (13x - 85)°. Find x.
Worked answer
17
13x - 85 = 2(4x)
5x = 85
x = 17
Tangents
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Two tangents from an external point have lengths 4x + 9 and 9x - 36. Find x.
Worked answer
9
4x + 9 = 9x - 36
x = 9
Both are 45 ✓
Problem 2
Two tangents from an external point have lengths 4x + 5 and 7x - 31. Find x.
Worked answer
12
4x + 5 = 7x - 31
x = 12
Both are 53 ✓