Circles — parts and theorems
The parts of a circle
- radius centre to edge; diameter right across ($= 2 \times$ radius); circumference the distance round.
- chord joins two points on the circle; tangent touches at just one point.
- arc part of the circumference; sector a slice between two radii; segment the region cut off by a chord.
Practice
Match each circle part to its meaning.
A chord joins two points; a tangent touches once; a sector is a slice between radii.
Circle theorems (both levels)
- The angle in a semicircle is $90^{\circ}$.
- The angle between a tangent and a radius is $90^{\circ}$.
Practice
The angle in a semicircle is how many degrees?
An angle standing on a diameter is always 90°.
Circle theorems (Extended)
- The angle at the centre is twice the angle at the circumference (same arc).
- Angles in the same segment are equal.
- Opposite angles of a cyclic quadrilateral add up to $180^{\circ}$.
- Worked example: circumference angle $40^{\circ}$ → centre angle $2 \times 40 = 80^{\circ}$.
Practice
The angle at the circumference is 40°. The angle at the centre on the same arc is how many degrees?
Angle at centre = 2 × angle at circumference = 2 × 40 = 80°.
Practice
In a cyclic quadrilateral one angle is 85°. Its opposite angle is how many degrees?
Opposite angles add to 180°: 180 − 85 = 95°.
You've got it
Key idea
- angle in a semicircle $=90^{\circ}$; tangent meets radius at $90^{\circ}$
- angle at centre $= 2\times$ angle at circumference (same arc)
- cyclic quadrilateral: opposite angles add to $180^{\circ}$