Lines and circles, angles at the centre and circumference, tangents
Slide a line across a circle and see when it cuts the circle, when it just touches it and when it misses. Bring two circles together, turn a tangent round a circle, and see that the angle at the circumference is always half the angle at the centre.
- UK: Year 10
- France: grade 8
- Germany: grade 5
- Iran: grade 8
- Afghanistan: grade 9
- Syria: grade 8
- Palestine: grade 9
- Ukraine: grade 7
- Russia: grade 7
Guide
For GCSE (Higher circle theorems) and Iran Grade 8 / Afghan Grade 9. The coloured dots and hollow rings can be dragged; the readouts under the figure update live.
Call the distance from the centre O to a line d and the radius r. If d > r the line misses the circle; if d = r it is a tangent and touches the circle at exactly one point; if d < r it cuts the circle at two points. A tangent is always perpendicular to the radius at the point of contact.
An angle at the centre has its vertex at the centre O. An angle at the circumference has its vertex on the circle, with two chords as its arms. The angle at the centre is twice the angle at the circumference standing on the same arc. The angle in a semicircle is always 90°.