Parts of a Circle
Theory
Every point on a circle is the same distance from the centre; that distance is the radius. A diameter is two radii end to end, \(d = 2r\), so a circle with diameter \(6\) cm has radius \(3\) cm. This Year 8 maths page names every part of a circle.
The centre is usually labelled \(O\) (or \(C\)). Every radius has the same length, so two radii and a chord make an isosceles triangle.
A diameter is a chord through the centre: \(d = 2r\) and \(r = \dfrac{d}{2}\). It is the longest chord, so no chord is longer than \(2r\).
A chord joins two points on the circle. An arc is part of the circle; the circumference is the whole way around. Two points cut the circle into a shorter minor arc and a longer major arc.
A sector is the region between two radii and an arc (a minor or major sector). A quadrant is a quarter-circle sector; a semicircle is half a circle, cut off by a diameter.
A segment is the region between a chord and an arc. A tangent is a line that touches the circle at exactly one point.
Pi. For every circle, circumference \(\div\) diameter is the same number, \(\pi\) (pi) \(\approx 3.14159\).
Circles with the same centre are concentric: the ring between them is the big radius minus the small radius wide. Circles that touch have centres the sum of the radii apart (touching outside) or the difference (one inside the other).
The parts of a circle:
| Part | Meaning |
|---|---|
| radius | centre to the circle |
| diameter | chord through the centre, \(d = 2r\) |
| chord | joins two points on the circle |
| arc | part of the circumference |
| circumference | the whole way around |
| sector | region between two radii and an arc |
| quadrant | a quarter-circle sector |
| semicircle | half a circle |
| segment | region between a chord and an arc |
| tangent | touches at exactly one point |
How to name a part
- Ask: is it a line, a curve or a region?
- Line: through the centre (diameter), from the centre (radius), between two points (chord) or touching once (tangent).
- Region: two radii (sector) or a chord (segment).
| \(CB\) | \(=\) | \(\dfrac{1}{2} \times 6 = 3\) |
\(CB\) is a radius, so \(CB = 3\) cm.
(i) Find the area of the minor sector \(ACB\).
(ii) Find the area of the minor segment cut off by the chord \(AB\).
| \(\text{sector}\) | \(=\) | \(\dfrac{1}{4} \times 16\pi = 4\pi\) |
| \(\triangle ACB\) | \(=\) | \(\dfrac{1}{2} \times 4 \times 4 = 8\) |
| \(\text{segment}\) | \(=\) | \(4\pi - 8\) |
(i) \(4\pi~\text{cm}^2\)
(ii) \((4\pi - 8)~\text{cm}^2\): the sector take away the triangle.
(i) a region bounded by two radii and an arc
(ii) an interval joining two points on the circle that misses \(O\)
(iii) a line that touches the circle once
(i) sector
(ii) chord (not a diameter, because it misses \(O\))
(iii) tangent
| \(r + r + 9\) | \(=\) | \(29\) |
| \(2r\) | \(=\) | \(20\) |
| \(r\) | \(=\) | \(10\) |
\(OP\) and \(OQ\) are both radii, so \(r = 10\) cm. Check: \(10 + 10 + 9 = 29\).
| \(d\) | \(=\) | \(9\) |
| \(r\) | \(=\) | \(9 \div 2 = 4.5\) |
The width of the courtyard is a diameter, so the radius is \(4.5\) m.
(i) Work out circumference \(\div\) diameter, correct to \(2\) decimal places.
(ii) What is that number called?
| \(\dfrac{138}{44}\) | \(\approx\) | \(3.14\) |
(i) \(3.14\)
(ii) \(\pi\), the same ratio for every circle (\(\pi = 3.14159\ldots\)).
Common pitfalls
Frequently asked questions
What is the difference between a radius and a diameter?
A radius goes from the centre to the circle; a diameter goes right across through the centre and is twice as long: \(d = 2r\).
What is a chord?
An interval joining two points on a circle.
What is a sector?
The region inside a circle between two radii and an arc, like a slice of pizza.
What is a segment of a circle?
The region between a chord and the arc it cuts off.
What is pi?
The circumference of any circle divided by its diameter, about \(3.14159\).