Circles — parts and theorems
Introduced| English | Meaning |
|---|---|
| radius/ˈreɪdɪəs/ | The distance from the centre of a circle to its edge. |
| diameter/daɪˈæmɪtə/ | A straight line through a circle's centre joining two points on its circumference, or the length of that line. |
| circumference/sɜːˈkʌmfrəns/ | circumference |
| chord/kɔːd/ | chord |
| tangent/ˈtændʒənt/ | A straight line that touches a curve at a point and has the same direction there. |
| arc/ɑːk/ | A continuous portion of a curved line, such as part of a circle's circumference. |
| sector/ˈsektə/ | A region of a circle bounded by two radii and the arc between them. |
| segment/ˈseɡmənt/ | A region of a circle bounded by a chord and the arc joining its endpoints. |
| cyclic quadrilateral/ˈsaɪklɪk ˌkwɒdrɪˈlætərəl/ | A quadrilateral whose four vertices lie on the same circle. |
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.
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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?
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An angle standing on a diameter is always 90°.
A tangent to a circle meets the radius at 90°.
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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?
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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?
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Opposite angles add to 180°: 180 − 85 = 95°.
Angles in the same segment are ______.
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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
The missing Extended circle facts
- Alternate segment theorem: the angle between a tangent and a chord equals the angle at the circumference in the opposite segment. If that tangent-chord angle is $52^{\circ}$, the matching circumference angle is $52^{\circ}$.
- The perpendicular bisector of a chord passes through the centre; equal chords are equally far from it. Tangents PA and PB from the same outside point P have equal lengths, so $PA=7$ cm implies $PB=7$ cm.
A tangent-chord angle is 52°. Find the angle in the alternate segment in degrees.
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The alternate segment theorem makes these two angles equal.
PA and PB are tangents from the same external point P. PA is 7 cm. Find PB in cm.
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Tangents from the same external point have equal lengths.
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 $180^{\circ}$
Write from a centre angle of 70
- The angle at the circumference is 35 degrees.
- The angle at the centre on the same arc is twice that.
The angle at the circumference is 35 degrees. What is the angle at the centre standing on the same arc?
2 times 35 is 70.
What is the angle at the centre?
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The centre angle is double. Other options — 80 (the lesson centre angle); 35 (the circumference angle left unchanged).