**Angles**

The measures of the angles are determined using the angles formulae. Two rays that intersect at a same point and are referred to as the angle’s arms are what make up an angle. The vertex of the angle is the name for the angle’s corner point. The amount of rotation between the two lines is referred to as the angle. Angles are measured in radians or degrees. Let’s talk about the circle’s central angle formulae, numerous angle formulas, and double angle formulas along with a few cases that have been solved.

**What Are Angle Formulas?**

The formulas for the angle created at the circle’s centre by two radii and an arc have been covered in this article. Let’s also concentrate on the trigonometric formulas for numerous angles and double angles.

**Central Angle Formula**

\text { Angle }=\frac{\text { Arc Length } \times 360^{\circ}}{2 \pi \text { Radius }}

**Formula for Central Angle**

s=r \theta

**Angle Definition**

The vertex, which is the common point where two lines or rays diverge, forms the angle. When two rays, or half-lines projected with a common termination, cross, an angle is created. The vertex of the angle is formed by the angle’s corner points, and its sides—i.e., the lines—are represented by the angle’s rays.

**Types of Angles:**

- Acute Angle – 0° to 90°, both exclusive.
- Obtuse Angle – 90°to 180°, both exclusive.
- Right Angle – Exactly 90°
- Straight Angle – Exactly 180°
- Reflex Angle – 180° to 360° both exclusive.
- Full Rotation – Exactly 360°

**Question 1**

**Determine the angle of a segment in a circle where the radius measures 8 units and the arc length is 3π.**

**Solution:**

We know that the formula for calculating the angle of a segment in a circle is:

θ = (l / r) * 180° / π

where θ is the angle in degrees, l is the length of the arc, and r is the radius of the circle.

Substituting the given values, we get:

θ = (5π / 6) * 180° / π

θ = 150°

Therefore, the angle of the segment in the circle is 150 degrees.

**Question 2**

**Given a circular segment with an arc length of 6π and a radius of 10 units, find the angle of the segment using the angle formulas.**

**Solution:**

The formula for calculating the angle of a circular segment is given by:

θ = (l / r) * (180 / π)

where θ is the angle in degrees, l is the length of the arc, and r is the radius of the circle.

Substituting the given values, we get:

θ = (7π / 9) * (180 / π)

θ = 140°

Therefore, the angle of the circular segment is 140 degrees.

**Question 3**

**Determine the angle of a segment formed in a circle with radius of 8 cm and arc length of 7π.**

**We can use the formula:**

angle (in radians) = arc length / radius

Plugging in the values, we get:

angle = 7π / 8

To simplify this expression, we can divide both the numerator and denominator by the greatest common factor of 7 and 8, which is 1:

angle = 7π / 8 = (7/1) * (π/8) = (7/8) * π

Therefore, the angle of the segment formed in the circle is (7/8)π radians or approximately 2.745 radians. To convert this to degrees, we can multiply by 180/π:

angle = (7/8)π * (180/π) = 157.5 degrees

**What Do Angle Formulas Mean?**The measures of the angles are determined using the angles formulae. A figure called an angle is made up of two intersecting rays that share a terminal. These rays are referred to as the angle’s arms. The vertex of the angle is the name for the angle’s corner point. The amount of rotation between the two lines is referred to as the angle. The formulas for angles cover a variety of issues, including the central angle of a circle and multiple angles.

**What are the Angle Calculation Formulas?**Angle formulas at a circle’s centre can be written as,

Central angle, = Arc length/r radians, where r is the circle’s radius, or Central angle, = (Arc length 360o)/(2r) degrees.

**Making use of the Angles Formula, With r = 8 units and an angle of /2, find the length of the arc.**The arc’s angle is 90 degrees.

A circle’s radius is equal to 8 units.

Using the formula for angles, = s/r

s = 8π/2

= 4π

As a result, a circle’s arc has a length of 4.