Circle and the radian measure

Terminology is important, because if you can’t understand the question, you can’t expect to earn marks.

Central angle
Arc length
The radian measure
Sector
Chord and secant
Segment
Tangent

Central angle

The central angle, $\theta$ , is subtended (formed) by two radii of the same circle. Because the center is often labeled $O$ , central angles are often denoted as $A\hat OB$ , with $A$ and $B$ on the circumference and the two radii.

$\theta$ can also refer to other angles, but on this page it’s the central angle.

Arc length

The arc, which is part of the circumference enclosed by two radii, is associated with an arc length.

l = \text{circumference}\cdot\frac{\theta}{360\degree} = \frac{\pi r\theta}{180\degree}

Note: the “arc measure” refers to the angle of an arc. The terms are not interchangeable.

The radian measure

The radian measure is an alternative unit to measure an angle. $1 \text{rad}$ is roughly $57.3\degree$ . It is defined by

\text{radian} = \frac{\text{arc length}}{\text{radius}}

Because the circumference is $C = 2\pi r$ ,

1\text{ radian} = \frac{1}{2\pi} \text{turn} = \frac{360\degree}{2\pi} = \frac{180\degree}{\pi}

The arc length in radians is simply

l = \theta r, \;\; 0 \leq \theta \leq 2\pi

The radian is denoted by $\text{rad}$ , $^c$ , or $^r$ and similar symbols. When there is neither a degree or a radian symbol, the angle is in radians.

Sector

A sector is a slice of a circle, formed by two radii.

The area of a sector is

A_\text{sector} = \frac{\text{angle in degrees}}{360\degree}\cdot \pi r^2

A_\text{sector} = \frac{\text{angle in radians}}{2\pi}\cdot \pi r^2 = \frac 12 \theta r^2

The perimeter of a sector is the arc length plus twice the radius.

Chord and secant

A chord is a line segment connecting two points on the circumference, and it is shorter than the diameter. The line containing the chord is the secant.

Segment

A chord splits the circle into two regions. The small region is called a minor segment, or simply segment.

The area of segment shall be discussed later, with introduction of the radian measure.

Tangent

A tangent is a line that is just touching the circle at one point. A tangent line is perpendicular to the radius at the point of tangency.

For any point outside the circle, two different tangent lines can be drawn. This point form an isosceles triangle with the two points of tangency.

two tangent lines — Two tangent lines forming an isosceles triangle with the chord connecting the two points of tangency