Circle Angles, Arcs & Chords — Print
Where the vertex of an angle sits — centre, edge or outside — decides which rule applies.
What the practice questions 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.
Three people stand on the edge of a circular field, all watching the same pair of goalposts.
They are in completely different places. Is the angle they see the goalposts at the same for each?
It is. Every point on that arc sees the goalposts at exactly the same angle.
Move to the centre instead and the angle doubles — the centre sees twice what the edge sees.
Step outside the circle and it shrinks again.
So there is no single "circle angle" rule. There are three, and which one you need depends entirely on where the vertex is: at the centre, on the circle, or outside it.
Vertex at the centre
A central angle is exactly equal to the arc it cuts. A 70° central angle cuts a 70° arc.
This is really a definition — arcs are measured by their central angles.
The whole circle is 360°, so a semicircular arc is 180° and a quarter is 90°.
Vertex on the circle
An inscribed angle is half its arc. The same 70° arc is seen as 35° from the edge.
Two consequences follow immediately. Angles on the same arc are congruent, because they halve the same number.
And an angle in a semicircle is a right angle, because half of 180° is 90°.
The one to remember
Centre sees the whole arc; edge sees half of it. The centre is always twice the edge for the same arc, and nearly every circle question uses that.
Vertex outside
Two secants, two tangents, or one of each meeting outside the circle make an angle equal to half the difference of the two arcs they cut.
If the far arc is 110° and the near arc is 40°, the angle is 110 - 402 = 35°.
Vertex inside but not at the centre gives half the sum instead.
Three positions, three rules
Centre: the arc. On the circle: half the arc. Inside: half the sum. Outside: half the difference. Find the vertex first and the rule chooses itself.
Chords
Congruent chords cut congruent arcs, and sit the same distance from the centre.
A radius perpendicular to a chord bisects it — and the converse holds too.
When two chords cross inside a circle, the products of the pieces are equal: a · b = c · d.
Tangents
A tangent is perpendicular to the radius at the point of contact. That right angle turns most tangent problems into Pythagoras.
Two tangents drawn from the same external point are congruent.
And a tangent-chord angle is half its intercepted arc — the inscribed angle rule, with one arm flattened onto the circle.
Worked examples
Inscribed angle
An inscribed angle cuts an arc of 86°. Find the angle, and the central angle on the same arc.
- Inscribed angle is half its arc. Vertex on the circle.
- 86 ÷ 2 = 43° The inscribed angle.
- Central angle equals its arc. Vertex at the centre.
- 86° — twice the inscribed angle, as it must be ✓ Same arc, so centre is double edge.
Inscribed 43°; central 86°
Angle from outside
Two secants from an external point cut arcs of 120° (far) and 46° (near). Find the angle at the point.
- Vertex outside, so half the difference. Not half the sum — that is for inside.
- 120 - 46 = 74 Far arc minus near arc.
- 74 ÷ 2 = 37° Half the difference.
- Sense check: it is smaller than the inscribed angle would be ✓ Stepping back always shrinks the angle.
37°
Tangent and radius
A tangent touches a circle of radius 9 at T. From external point P, PO = 15 where O is the centre. Find PT.
- The tangent is perpendicular to the radius at T. So △ OTP has a right angle at T.
- OT = 9 (radius) and OP = 15 (hypotenuse). The hypotenuse is opposite the right angle.
- PT2 = 152 - 92 = 225 - 81 = 144 Pythagoras, subtracting to find a leg.
- PT = 12 A 9-12-15 triangle — a 3-4-5 tripled ✓
PT = 12
Words to know
- 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.
- Arc
- Part of the circle, measured in degrees like the angle that cuts it.
- Secant
- A line that cuts the circle at two points.
Mistakes to watch for
Using the inscribed rule when the vertex is at the centre. A central angle equals its arc; only an angle with its vertex on the circle is halved. Locate the vertex before choosing a rule.
Taking half the sum when the vertex is outside. Inside the circle it is half the sum; outside it is half the difference. Getting these backwards gives an answer that is often larger than the whole arc.
Forgetting that a tangent meets the radius at a right angle. That right angle is what makes Pythagoras available, and without it these problems look unsolvable.
Assuming a chord passes through the centre. Only a diameter does. A radius bisects a chord only when it is perpendicular to it, and that has to be given or proved.
That every angle on the same arc is equal is called Thales' theorem in its most famous special case: any angle drawn in a semicircle is a right angle.
Thales is said to have sacrificed an ox in celebration when he proved it.
The result is why a carpenter can find the centre of a circular tabletop with nothing but a set square.
Problem 1
A central angle cuts an arc of 70°. Find the angle.
Worked answer
70
70°
Problem 2
An inscribed angle cuts an arc of 86°. Find the angle.
Worked answer
43
86 ÷ 2 = 43°
Problem 3
An angle is inscribed in a semicircle. Find its measure.
Worked answer
90
The arc is 180°.
Half is 90°.
Problem 4
An inscribed angle measures 35°. Find its arc.
Worked answer
70
2 × 35 = 70°
Problem 5
An angle with its vertex at the centre equals
- its arc
- half its arc
- twice its arc
- half the difference
Worked answer
its arc
At the centre, so it equals the arc.
Problem 6
Two secants from an external point cut arcs 120° and 46°. Find the angle.
Worked answer
37
120 - 46 = 74
74 ÷ 2 = 37°
Problem 7
Two chords cross inside a circle cutting arcs 80° and 40°. Find the angle.
Worked answer
60
80 + 40 = 120
120 ÷ 2 = 60°
Problem 8
A tangent touches a circle of radius 9 at T. From P, OP = 15. Find PT.
Worked answer
12
PT2 = 152 - 92 = 144
PT = 12
Problem 9
Two chords cross so the pieces are 3, 8 and 4, x. Find x.
Worked answer
6
3 × 8 = 4x
24 = 4x
x = 6
Problem 10
Two inscribed angles stand on the same arc. One is 28°. Find the other.
Worked answer
28
Angles on the same arc are congruent.
28°
Problem 11
A tangent meets the radius at the point of contact at
- 90°
- 45°
- 180°
- any angle
Worked answer
90°
A tangent is perpendicular to the radius there.
Problem 12
A central angle is 124°. Find the inscribed angle on the same arc.
Worked answer
62
124 ÷ 2 = 62°
Problem 13
In a cyclic quadrilateral one angle is 105°. Find the opposite angle.
Worked answer
75
180 - 105 = 75°
Problem 14
Two tangents from an external point cut arcs of 210° and 150°. Find the angle at the point.
Worked answer
30
210 - 150 = 60
60 ÷ 2 = 30°
Problem 15
An inscribed angle is (3x)° and its arc is (8x - 20)°. Find x.
Worked answer
10
8x - 20 = 2(3x)
8x - 20 = 6x
x = 10
Problem 16
Two tangents are drawn from an external point. One has length 3x + 2, the other 5x - 8. Find x.
Worked answer
5
3x + 2 = 5x - 8
10 = 2x
x = 5