Circle Angles, Arcs & Chords

Where the vertex of an angle sits — centre, edge or outside — decides which rule applies.

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?

three people watching the same goalposts A B C they stand in different places — is the angle the same?

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.

where the vertex sits decides the rule centre = the arc on the circle = half the arc outside = half the difference the orange arc is the same in all three pictures; only the vertex moved

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°.

A 70 degree central angle cuts a 70 degree arc; a semicircle is 180 degrees and a quarter circle 90 a central angle and its arc are the same number 70° O 70° arc 180° 90° semicircle180° arc quarter circle90° arc arcs are measured bytheir central angles; the whole circle is 360°

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°.

An inscribed angle of 35 degrees and a central angle of 70 degrees standing on the same 70 degree arc, and a right angle in a semicircle the same arc: 70° from the centre, 35° from the edge 70° 35° O P 70° arc angle in a semicircle 90° 180° arc half of 180° = 90° the centre is twice the edge

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.

Two secants from an outside point cut arcs of 110 and 40 degrees and meet at 35 degrees; two chords crossing inside meet at half the sum outside: half the difference 35° P 110°40° (110° − 40°) ÷ 2 = 35° inside: half the sum 70° 100°40° (100° + 40°) ÷ 2 = 70°

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.

Two chords crossing inside a circle: pieces 4 and 6 on one, 3 and 8 on the other, and 4 × 6 = 3 × 8 two chords crossing: the products of the pieces match a = 4b = 6 c = 3d = 8 a · b = c · d: 4 × 6 = 3 × 8 = 24 radius ⟂ chord it cuts the chordinto two equal halves

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.

Two tangents from an outside point P: each meets its radius at a right angle, and PA = PB; with radius 5 and OP 13, PA is 12 OP AB 51213 tangent ⟂ radius at the point of contact PA = PB two tangents fromone outside point 5² + 12² = 13² the right angle turnsit into Pythagoras

Worked examples

Inscribed angle

An inscribed angle cuts an arc of 86°. Find the angle, and the central angle on the same arc.

  1. Inscribed angle is half its arc. Vertex on the circle.
  2. 86 ÷ 2 = 43° The inscribed angle.
  3. Central angle equals its arc. Vertex at the centre.
  4. 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.

  1. Vertex outside, so half the difference. Not half the sum — that is for inside.
  2. 120 - 46 = 74 Far arc minus near arc.
  3. 74 ÷ 2 = 37° Half the difference.
  4. 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.

  1. The tangent is perpendicular to the radius at T. So △ OTP has a right angle at T.
  2. OT = 9 (radius) and OP = 15 (hypotenuse). The hypotenuse is opposite the right angle.
  3. PT2 = 152 - 92 = 225 - 81 = 144 Pythagoras, subtracting to find a leg.
  4. 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.

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