Where the vertex sits is everything.
central angle
vertex at the centre
equals its arc exactly
inscribed angle
vertex on the circle
half its arc
- Home
- Lesson Library
- Central and inscribed angles
Central and Inscribed Angles
An angle at the centre equals its arc. An angle on the rim is half of it.
Find the arc
An inscribed angle measures 35°. How big is its arc?
An inscribed angle is half the arc it stands on, so the arc is 2 × 35 = 70°. D halves instead of doubling. Remember which is the small one: the angle sits at the edge, the arc wraps around.
From an inscribed angle to its arc.
inscribed = ½ × arc
the rule
35 = ½ × arc
substitute
arc = 70°
double it
Which way round?
The arc is always the bigger number. Going from the angle to the arc you double; from the arc to the angle you halve. Halving 35 to get 17.5 is the standard slip.
Central to inscribed, on the same arc.
central 100° → arc = 100°
they are the same number
inscribed = ½ × 100
half the arc
= 50°
done
Every inscribed angle standing on the same arc is the same size, wherever on the rim its vertex sits. Only the arc matters.
The angle in a semicircle is 90°.
A special case of the halving rule, and it comes up constantly.
a diameter cuts a 180° arc
half the circle
inscribed = ½ × 180
the same rule
= 90°
always a right angle
So any triangle drawn with the diameter as one side is right-angled — and Pythagoras is suddenly available.
Four-sided figures inside a circle.
Opposite angles of a cyclic quadrilateral add to 180°.
| Fact | Rule |
|---|---|
| central angle | = its arc |
| inscribed angle | = half its arc |
| angles on the same arc | equal to each other |
| angle in a semicircle | 90° |
| opposite angles, cyclic quad | add to 180° |
| all arcs round the circle | add to 360° |
Given one angle of a cyclic quadrilateral as 85°, the opposite one is 95° without any further work.
Variations: the vertex on the small arc, an arc made of two pieces.
O is the centre, ∠AOB = 110°, and C lies on the minor arc AB. Find ∠ACB.
minor arc AB = 110°
C sits on this one
∠ACB stands on the major arc = 360 − 110 = 250°
the arc opposite the vertex
∠ACB = ½ × 250 = 125°
an obtuse inscribed angle
The angle stands on the arc it does not touch
½ × 110 = 55° is the angle when C is on the major arc. With C on the minor arc the answer is 180 − 55 = 125°. The two positions together make a cyclic quadrilateral, so they always total 180°.
ABCD is a cyclic quadrilateral with arc BC = 84° and arc CD = 120° (neither contains A). Find ∠BAD and ∠BCD.
∠BAD stands on arc BCD = 84 + 120 = 204°
add the two pieces
∠BAD = ½ × 204 = 102°
half its arc
∠BCD = 180 − 102 = 78°
or ½(360 − 204) = 78 ✓
Different: the arc under the angle is split by a vertex. Follow the circle from one arm to the other, away from the vertex, and add every piece on the way.
When one side of the angle is a tangent.
A tangent touches the circle at A, and chord AB cuts off an arc of 130° on one side. Find the angle between the tangent and the chord on that side.
tangent–chord angle = ½ × its arc
it behaves like an inscribed angle
= ½ × 130
the arc inside the angle
= 65°
the other side is 180 − 65 = 115°
Why it works: the radius OA meets the tangent at 90°, and the central angle AOB is 130°, so in isosceles △OAB each base angle is 25°. The tangent–chord angle is 90 − 25 = 65° — half the arc again.
Two companion facts: a radius drawn to the point of contact is perpendicular to the tangent, and the two tangent segments from one outside point are equal. The first turns many circle figures into right triangles.
On the exam
Circle-angle items chain two rules: a diameter gives 90° and then the triangle sum finishes it; two angles on the same arc set equal; a cyclic quadrilateral gives 180° and an isosceles triangle splits the rest. Mark the arc each angle stands on before doubling or halving. Ten-second check: an angle on the rim is always half the angle at the centre on the same arc.
Check: do the arcs close the circle?
our arc
70°
from the 35° angle
the rest
360 − 70 = 290°
the major arc
sensible?
35 < 70 < 360
angle smaller than its arc ✓
Now you try
Practice problem
Find the inscribed angle
An arc measures 128°. Find the inscribed angle standing on it.
Half the arc: 128 ÷ 2 = 64°. B doubles instead. A central angle would equal the arc exactly — only the inscribed angle halves it, and that is the distinction the question turns on.