Arcs and central angles
A circle is the set of all points at a fixed distance from a centre. Its parts and angles follow elegant rules used in engineering, navigation, and design.
Parts of a circle
A circle is named by its centre, such as circle O. The radius is the distance from the centre to the circle; the diameter, twice the radius, passes through the centre. A chord joins two points on the circle, and the diameter is the longest chord. A secant line cuts the circle twice; a tangent touches it at exactly one point.
Circumference and pi
The circumference, the distance around a circle, is \(C = 2\pi r = \pi d\). The constant \(\pi \approx 3.14159\) is the ratio of any circle's circumference to its diameter. Archimedes, around 250 BCE, showed it lies between 223/71 and 22/7 by trapping a circle between polygons with 96 sides.
Central angles
A central angle has its vertex at the centre of the circle. The angles around the centre add to 360 degrees, so a circle divided into 8 equal slices has central angles of 45 degrees. Pie charts use central angles to show proportions: a category that is 25% of the data gets a 90-degree slice.
Arcs and their measures
An arc is part of a circle. A minor arc is less than half the circle and has the same measure as its central angle; a major arc is more than half, measuring 360 minus the minor arc; a semicircle measures 180 degrees. Minor arcs are named with two letters, such as arc AB, and major arcs with three, such as arc ACB.
Arc addition
Adjacent arcs add like adjacent angles: if arcs AB and BC share only point B, then the measure of arc ABC equals arc AB plus arc BC. In the same circle, or in congruent circles, congruent central angles cut off congruent arcs and congruent chords.
Arc length
Arc length is a fraction of the circumference: \(L = \frac{\theta}{360} \times 2\pi r\). An arc of 60 degrees on a circle of radius 9 cm has length \(\frac{60}{360} \times 18\pi = 3\pi \approx 9.42\) cm. Arc measure is in degrees; arc length is a distance.
Radians
Radians measure angles by arc length: an angle of 1 radian cuts off an arc equal to the radius. A full turn is \(2\pi\) radians, so \(180° = \pi\) radians and \(90° = \tfrac{\pi}{2}\). In radians, arc length is simply \(L = r\theta\), which is why calculus and physics prefer radians.
Worked example
A bicycle wheel has a diameter of 70 cm. Each full turn moves the bike \(70\pi \approx 219.9\) cm. To travel 1 km, or 100,000 cm, the wheel must turn about 100,000 / 219.9, roughly 455 times. Turning through 120 degrees moves the bike one third of a circumference, about 73.3 cm.
Key takeaways
A circle has a centre, radius, and diameter, with chords, secants, and tangents meeting it. Circumference is 2 pi r, and pi is the circumference-to-diameter ratio, which Archimedes bounded around 250 BCE. Central angles at the centre sum to 360 degrees. A minor arc equals its central angle, a major arc is 360 minus the minor arc, and adjacent arcs add. Arc length is the angle's fraction of the circumference. In radians, 180 degrees is pi and arc length is r times theta.
Flashcard
Centre to circle distance
Check your understanding
What is the length of a 60-degree arc on a circle of radius 9 cm?