## Central angle. Arc of a circle.

A circle is a set of points in a plane that are equidistant from a fixed point called the center. The circle is an important shape in mathematics, and it is used in many fields, including geometry, trigonometry, and calculus.

**Central angle** is an angle whose vertex is the center of the circle. It is formed by two radii of the circle that connect the center to two points on the circle. In other words, a central angle is an angle whose vertex is at the center of a circle and whose arms intersect two points on the circle.

Consider a circle with center O, and let A and B be two points on the circle. The central angle \( \angle AOB \) is the angle formed by the two radii of the circle that intersect A and B at points O, A, and B.

The measure of a central angle \(^1\) is defined as the angle that it intercepts on the circumference of the circle, and is equal to the ratio of the length of the intercepted arc to the radius of the circle. We can express this relationship mathematically as: $$ \small \text{Measure of central angle } \angle AOB =\frac{ \text{Length of intercepted arc AB}}{ \text{Radius of circle}} $$

We can also express this formula in terms of the central angle's degree measure. Since the circumference of a circle is given by \( 2 \pi r \), where \(r\) is the radius of the circle, and there are 360 degrees in a full circle, we have:

\( \text{Length of intercepted arc AB } = \frac{\theta }{360^\circ} (2 \pi r) \) , here \( \theta \) is the degree measure of the central angle. Substituting this into the formula for the measure of the central angle, we get: $$ \small \text{Measure of central angle} \angle AOB = \frac{\theta }{360^\circ}(2r) = \frac{ \theta }{180^\circ} r $$

This formula is particularly useful when we know the radius of the circle and the degree measure of the central angle, and we want to find the length of the intercepted arc or the measure of the angle that subtends the arc.

The measure of a central angle \(^2\) is equal to the measure of the arc it intercepts. This relationship can be expressed mathematically as: \(\theta = \frac{s}{r} \), where \(\theta \) is the measure of the central angle in radians, \(s\) is the length of the arc intercepted by the angle, and \(r\) is the radius of the circle.

For example, if the radius of a circle is \(r=5\) and an arc of length \(s=3\) intercepts a central angle, the measure of the angle can be found using the formula: \(\theta = \frac{s}{r}=\frac{3}{5} \)

So the measure of the central angle is \( \theta =0.6 \text{radians} \).

**Arc of a circle** is a portion of the circumference of a circle. It is defined by two endpoints on the circle and is the shortest path between them. The length of an arc can be found using the formula: \(s = r \theta \) , where \(s\) is the length of the arc, \(r\) is the radius of the circle, and \(\theta \) is the measure of the central angle in radians.

For example, if the radius of a circle is \(r=2\) and the central angle intercepts an arc of length \(s=3\), the measure of the angle can be found using the formula: \(\theta = \frac{s}{r}=\frac{3}{2} \)

So the measure of the central angle is \( \theta = 1.5 \) radians, and the length of the arc is: \(s = r \theta = 2 \cdot 1.5 =3 \)

Thus, the arc has a length of 3 units.

Let's consider a circle with center \(O\) and radius \(r\). Suppose that we have two points \(A\) and \(B\) on the circle such that \(A\) and \(B\) are not diametrically opposite points (that is, they do not lie on a line passing through the center of the circle). The arc of the circle that is intercepted by these two points is the portion of the circle's circumference that lies between \(A\) and \(B\), including \(A\) and \(B\) themselves.

The length of an arc of a circle is given by the formula: \( \text{Length of arc } AB = \frac{ \theta }{360^\circ } (2 \pi r) \) , where \( \theta \) is the degree measure of the central angle that subtends the arc AB. This formula follows from the fact that the ratio of the arc length to the circumference of the circle is equal to the ratio of the angle that the arc subtends to the full angle of the circle (which is 360 degrees).

Alternatively, we can rearrange the formula to find the degree measure of the central angle that subtends an arc of length s on a circle with radius \(r\): $$ \text{Degree measure of central angle} = \frac{s}{r} \cdot \frac{180^\circ}{ \pi } $$

in addition to length, arcs of circles can also be measured in terms of their angle measure, which is the degree measure of the central angle that subtends the arc. If we know the radius of the circle and the angle measure of the central angle that subtends an arc, we can find the length of the arc using the formula above.

It is important to note that there are two types of arcs on a circle: **minor arcs** and **major arcs**. A minor arc is an arc that measures less than 180 degrees, while a major arc is an arc that measures greater than 180 degrees. A **semicircle** is a special case of a major arc that measures exactly 180 degrees.