Formula for Angles

Do you often come across various types of “angles” and how does one exactly define what an angle is? And why are there so many types of angles? If yes, then Vedantu is here to answer all the questions. Lest’s start by answering the question of what is an angle?

The angle is a very important measurement in geometrical shapes. The amount of rotation about the point of intersection of two lines or planes which is required to bring one in correspondence with the other is known as an Angle. There are many different types of angles in geometry. In this article, we will discuss the concepts of angles with a formula for angles and their examples. Let us begin learning.  

Angle Meaning

Let’s know the angle meaning. The angle is a shape, formed by two lines or rays diverging from a common point which is the vertex. The angle is formed when two rays intersect i.e. half-lines projected with a common endpoint. The corner points of angle are the vertex of the angle and the rays as the sides, i.e. the lines are known as the arms.  

We define it as the measure due to the turn between the two lines. The unit of angle in radians or degrees. There are different types of formulas for angles, some of them are a double-angle formula, half-angle formula, compound angle formula, interior angle formula, etc.  

It can be represented by three letters of the shape which define the angle, and the middle letter is where the angle is made. For example,   

∠ABC, where B is the given angle, made by lines AB and BC.  

The definition of a radian is the angle of an arc in a circle which is created by enclosing the radius of the circle around its circumference. We represent this angle between two lines through radians as well as degrees. The total angle of a circle equals 360\degree or we can call it as 2 radians. With the help of a formula for conversion from radians to degrees, we can convert the angles that are in radians into degrees. Therefore, we measure angle generally with the terms which are degree (°),  gradins or radians.

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Different Types of Angles in Geometry

• Acute Angle – 0° to 90°, both exclusive. Have you observed the angle between the cuts on a pizza? Don’t you think that an acute angle looks just similar to an acute angle? An angle that is less than a right angle and which measures less than 90 degrees is known as an acute angle. Acute angles are the angles that lie between 0 degrees and 90 degrees. In the figure given below, you can see the acute angle with a measure less than 90 degrees.

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• Obtuse Angle – 90°to 180°, both exclusive. An angle that measures more than a right angle that is more than 90 degrees and which measures less than 180° is obtuse. The obtuse angle lies between 90 degrees and 180 degrees. A door when kept wide is an example of an obtuse angle as when open, it forms an obtuse angle. In the figure given below, you can see the obtuse angle with a measure of more than 90 degrees.

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• Right Angle – Exactly 90°. You might find right angles anywhere you look around.  When you look at the houses, buildings and other structures incorporate the right angle in their construction? What is the right angle? Well, the angles that measure equal to 90 degrees are referred to as the right angles. In the figure given below, you can see the angle with measure 90°and is represented with a square-like box.

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• Straight Angle – Exactly 180°

• Reflex Angle – 180° to 360° both exclusive.

• Full Rotation – Exactly 360°

Other Different Types of Angles in Geometry

• Reflex Angle: A reflex angle can be defined as an angle that measures more than 180° but less than 360°.

• Straight Angle: The straight angle is an angle whose measure is equal to 180 degrees. Its straight angle looks like a straight line. They are collinear and opposite rays. When any thin book is kept open, we can see that the angle formed between the two pages is an example of a straight angle that is 180 degrees.

• Adjacent Angle: Two angles are said adjacent when they share a common side and a vertex.

• Vertically Opposite Angles: Vertically opposite angles are known to be the angles formed opposite to each other when two lines intersect.

• Complementary Angles: Two angles are known as complementary when they add up to 90°

• Supplementary Angles: Two angles are known as supplementary when adding up to 180°

The Formula for Angles

(1) Formula for Central Angle:  

This formula in a circle is as follows:

θ = Arc Length×180°π×r  

Where θ is the Angle, Arc Length is referred to the length of the arc, r is the length of the radius and the value of the constant π and r is the radius.  

(2) Central Angle Formula: s=r×θ

Wheres represents the arc length, θ is the central angle in radians and r is the length of the radius.

(3) Double Angle Formula: 

This has three main formulas as below:  

cos2θ= cos²θ−sin²θ  

sin2θ=2sinθcosθ  

tan2θ=2tanθ/1−2 tan²θ  

where θ  is the angle in a right-angled triangle.

Questions to be Solved 

Question 1

There are three angles formed at a point. If one of the angles is the right angle and measures 90 degrees, the second angle is the straight angle, then the third angle must be

• Acute angle

• Obtuse angle

• Right angle

• Straight angle

Answer:

The correct option is C. One of the angles is a right angle, that is it is equal to 90°, the second angle is a straight angle = 180°. As we know the sum of the angles at a point or that complete one rotation is equal to 360°Therefore, The third angle is equal to 360° – (180 + 90) = 90° =equals 90° which is a right angle.  

Conclusion

With this, you have completed different types of angles and their respective formulas. Studying this topic thoroughly has prepared you to proceed with other advanced questions in the field of geometry, not just in this standard but also of higher standards. As students, you are expected to remember these angles and their properties for a very long time as they will be used time and again. To ensure you remember these permanently you have to solve questions. You can start with the above-given questions themselves.