Circles
Chords, the angle at the centre, semicircles and cyclic quadrilaterals.
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The angle at the centre is double the angle at the edge.
Look: The same chord seen from the centre (2x) and from the edge (x).
✏️ Your turn!
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1Angle in a semicircle?
2Centre angle 140°. Edge angle?
3Edge angle 35°. Centre angle?
! Careful!
✗Doubling to get the edge angle.
✓The edge angle is half.
✗Thinking the angle in a semicircle varies.
✓It is always 90°.
See the whole lesson on one page
The angle at the centre is double the angle at the edge.
Pick two points on a circle. The angle they make at the centre is twice the angle they make at any point on the circle. A line from the centre to the middle of a chord meets it at 90°.
Think of it like thisLook at the same two seats on a giant wheel from the middle, then from the rim. From the middle the gap looks twice as wide.
Solved together
A chord makes 110° at the centre. What angle does it make at the edge?
The edge angle is half the centre angle.
110 ÷ 2 = 55
Answer55°.
For older learners and grown-ups: a little more
A chord 16 cm long in a circle of radius 10 cm: the perpendicular from the centre halves it (8 cm), so the distance d satisfies d² + 8² = 10², d = 6 cm. Opposite angles of a cyclic quadrilateral add to 180°.
Now practise it
Kanakku makes new questions on circles every time. It shows the working and gives a hint when you are stuck.