Circles — parts and theorems
| English | Chinese | Pinyin |
|---|---|---|
| radius | 半径 | bàn jìng |
| diameter | 直径 | zhí jìng |
| circumference | 圆周 | yuán zhōu |
| chord | 弦 | xián |
| tangent | 切线 | qiè xiàn |
| arc | 弧 | hú |
| sector | 扇形 | shàn xíng |
| segment | 弓形 | gōng xíng |
| cyclic quadrilateral | 圆内接四边形 | yuán nèi jiē sì biān xíng |
The wheel that changed the world
- The circle is the most perfect shape in nature — every point on its edge is the same distance from the centre.
- Ancient mathematicians discovered remarkable properties hidden in this simple shape.
The parts of a circle
- Radius 半径: centre to edge. Diameter 直径: right across ($= 2r$). Circumference 圆周: the distance round.
- Chord 弦: a line joining two points on the circle. Tangent 切线: touches at exactly one point.
- Arc 弧: part of the circumference. Sector 扇形: a slice between two radii. Segment 弓形: the region cut off by a chord.
A diameter is a special chord — the longest possible one, passing through the centre.
Arc and sector
Drag the angle and radius to see the arc (part of the circumference) and the sector (pie slice) it cuts off.
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 (Core)
- The angle in a semicircle is $90^{\circ}$ (an angle standing on a diameter).
- The angle between a tangent and a radius is $90^{\circ}$.
The main parts of a circle; a chord joins two points, a tangent touches at just one
The angle in a semicircle is how many degrees?
An angle standing on a diameter is always 90°.
A tangent to a circle meets the radius at 90°.
The tangent is always perpendicular to the radius at the point of contact.
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 圆内接四边形 add up to $180^{\circ}$.
The angle at the centre is always twice the angle at the circumference for the same arc.
Same arc only. The "angle at centre = 2 × angle at circumference" rule only works when both angles subtend the same arc. Different arcs don't follow this rule.
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°.
In a cyclic quadrilateral one angle is 85°. Its opposite angle is how many degrees?
Opposite angles add to 180°: 180 − 85 = 95°.
Angles in the same segment are ______.
Any two angles subtended by the same arc in the same segment are equal.
Worked example
- Angle at circumference $= 40^{\circ}$ → angle at centre $= 2 \times 40 = 80^{\circ}$.
- In a cyclic quadrilateral, one angle $= 85^{\circ}$ → opposite angle $= 180 - 85 = 95^{\circ}$.
A sector is a slice between two radii; a segment is cut off by a chord; an arc is part of the circumference
You've got it
- 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 $180^{\circ}$