Angles are formed when two lines intersect or meet at a point. It can also be defined as the measure of turn between two lines. Angle is measured in degrees or radians. The angles could be of different types.

- Central circle
- Tangent Circle
- Unit Circle
- Great Circle

The angle formed by the intersection of two secants, two tangents, or one tangent or one secant. In geometry, the tangent of a circle is the straight line that touches circle exactly at a single point and it never enters the interior of the circle. This is a thin line passing through infinitely close points over the circle. The application of tangent circle formula is various theorems or they are used for geometrical constructions or proofs too.

- It will touch the circle exactly at a single point only.
- It intersects the circle radius at the right angle.

\[\large \huge \left(y-y_{0}\right)=m_{tgt}\left(x-x_{0}\right)\]

The great circle is the largest one drawn over the sphere surface. The minimum distance between two points on the surface of the sphere would be marked as the distance of great circle. Traditionally, the great circle was popular as Romanian circle. The diameter of a sphere will coincide with the diameter of the great circle. It is used for navigation of large ships or aircraft.

Great circle formula = d = r cos ^{-1} [Cos δ_{1} Cos δ_{2} cos(λ_{1} – λ_{2}) + sin δ_{1} sin δ_{2}]

Where,

r is the radius of the earth

δ is the latitude

λ is the longitude

**Question 1: **Find the great circle distance if the radius is 4.7 km, latitude is (45^{o}, 32^{o}) and longitude is (24^{o}, 17^{o}) ?

**Solution:**

Given,

\[\large \sigma_{1},\sigma_{2}=45^{\circ},32^{\circ}\]

\[\large \Lambda_{1},\Lambda_{2}=24^{\circ},17^{\circ}\]

r=4.7 km

r= 4700 m

Using the above given formula,

\[\large d=4700\;cos^{-1}(0.52\times 0.83\times 0.75)+(0.85 \times 0.32)\]

\[\large d=4700\times 0.99\]

D = 4653 m

Any circle having radius one is termed as unit circle in mathematics. They are useful in trigonometry where the unit circle is the circle whose radius is centered at the origin (0,0) in the Euclidean plane of the Cartesian coordinate system. The example of a unit circle is given below in the diagram –

The general equation of circle is given below:

\[\large \left(x-h\right)^{2}+\left(y-k\right)^{2}=r^{2}\]

Where (h, k) are center coordinates and r is the radius.

The unit circle formula is:

\[\large x^{2}+y^{2}=1\]

Where x and y are the coordinate values.

**Question: **Show that the point \[\large P\left[\frac{\sqrt{3}}{3},\,\frac{\sqrt{2}}{\sqrt{3}}\right]\] is on the unit circle.

**Solution:**

We need to show that this point satisfies the equation of the unit circle, that is: \[\large x^{2}+y^{2}=1\]

\[\large \left[\frac{\sqrt{3}}{3}\right]^{2}+\left[\frac{\sqrt{2}}{\sqrt{3}}\right]^{2}\]

\[\large =\frac{3}{9}+\frac{2}{3}\]

\[\large =\frac{1}{3}+\frac{2}{3}\]

= 1

Therefore P is on the unit circle.

A central angle is formed between two radii of a circle where two points intersect and form a segment, and the distance between points is the arc length that is denoted by l in geometry.

A central formed at the center of the circle where two radii meet or intersect. The next term that justifies the definition of a central angle is the vertex. A vertex is a point where two points meet to form an angle. The vertex for a central angle would always be the central point of the circle.

\[\LARGE Central\;Angle\;\theta =\frac{Arc\;Length\times 360}{2\pi r}\]

**Example 1:** Find the central angle, where the arc length measurement is about 20 cm and the length of the radius measures 10 cm?

**Solution:**

Given r = 10 cm

Arc length = 20 cm

The formula of central angle is,

Central Angle θ = \[\LARGE \frac{Arc Length \times 360}{2 \times\pi \times r}\]

Central Angle θ = \[\LARGE \frac{20 \times 360}{2 \times 3.14 \times 10}\]

Central Angle θ = \[\LARGE \frac{7200}{62.8}\] = 114.64°

**Example 2:** If the central angle of a circle is 82.4° and the arc length formed is 23 cm then find out the radius of the circle.

**Solution:**

Given Arc length = 23 cm

The formula of central angle is,

Central Angle θ = \[\LARGE \frac{Arc\;Length \times 360}{2\times\pi \times r}\]

82.4° =\[\LARGE \frac{23 \times 360}{2\times\pi \times r}\]

The central angle is shown more clearly in the diagram with its formula. This is important to discuss other vertexes too because when two radii meet, there could be more angles like the convex central angle, and reflex angle. If the central angle is measured less than 180-degree then it is a convex central angle. If the central angle is measured more than 180-degree then it is a reflex central angle.

With this discussion, you have a clear understanding of different angles of a circle and their formulas. You just have to put the values in formulas and calculate the angle for real word problems too.