The circle has centre \(O\).

• \(OA\), \(OB\), and \(OC\) are radii and \(AB\) is a diameter.
• \(AC\) is a part of the circumference and it's called an arc.
• \(OAC\) is a sector of the circle.
• \(\angle AOC\) is a sector angle or central angle.

\(\theta\) is said to be the angle subtended at the centre of the circle by arc \(AC\).

There is a relationship between the size of a central angle and the arc length that it is subtended by.

The larger the central angle, the longer the arc length; the longer the arc length, the larger the central angle subtended by it.

To this point, most students may have only experienced angles that have been measured in degrees.

If the arc length is \(\dfrac{17}{360}\) of the circumference of the circle, then the angle subtended at the centre of the circle by that arc measures \(17^{\circ}\).

Angles can be measured in units other than degrees.

Gradians are not widely used, but there are some instances where they are still used in areas like surveying.

On the diagram, an angle of \(100\) gon is shown.

In degrees, we know this is a right angle measuring \(90^{\circ}\).

Gradians are not our concern in this module or course.

In other words, in a circle with radius \(r\), the arc length, which subtends an angle of \(1\) radian, is \(r\).

The unit abbreviation for radians is rad.

In the diagram, a circle with radius \(r\) and center \(O\) is shown.

Let arc length \(AB = OA = OB = r\), the radius of the circle.

By definition, \(\angle AOB = 1\) radian \(= 1\) rad. Extend the arc from \(B\) to \(C\) so that \(BC =\) arc length \(AB = r\).

Now, arc length \(AC = 2r\) and it follows that \(\angle AOC = 2\) rad.

Next, extend the arc from \(C\) to \(D\) so that the arc length

\(CD =\) arc length \(BC=r\).

Now, arc length \(AD = 3r\) and it follows that \(\angle AOD = 3\) rad.

To determine the number of radii that make up a specific arc length, divide the arc length by the length of the radius.

This would also be the size of the subtended angle in radians.

That is, if \(\theta\) is a central angle subtended by an arc of a circle, then \(\theta = \dfrac{\mbox{arc length}}{\mbox{radius}}\) and \(\theta\) is measured in radians.

What is the relationship between radians and degrees?

We know that for \(\theta\), in radians, \(\theta = \dfrac{\mbox{arc length}}{\mbox{radius}}\).

For a circle with radius \(r\), we also know that the circumference is \(2 \pi r\).

So how many radians are there in a complete revolution?

Using the formula, with arc length \(= 2 \pi r\) and radius \(r \gt 0\), \(\theta = \dfrac{2 \pi r}{r} = 2 \pi \) rad.

But we also know that a complete revolution is \(360^{\circ}\). So,

Dividing both sides of \((1)\) by \(2\), we obtain the relation

 Dividing both sides of equation \((2)\) by \(\pi\), we can estimate the size of \(1\) rad in terms of degrees

\(1\) rad \(= \dfrac{180^{\circ}}{\pi} \approx 57.3^{\circ}\)

Dividing both sides of equation \((2)\) by \(180\), we can estimate the size of \(1^{\circ}\) in terms of radians

\(1^{\circ} = \dfrac{\pi}{180}\) rad \(\approx 0.017\) rad