Formula for Angles

The angle is a very important measurement in the 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 being where the angle actually 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 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 the acute angles? An angle which is less than a right angle and which actually measures less than 90 degree 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 which measures more than a right angle that is more than 90 degrees and which measures less than 180° is an obtuse angle. The obtuse angle lies between 90 degrees and 180 degrees. A door when kept wide is an example of obtuse angle as when open, it forms an obtuse angle. In the figure given below, you can see that the obtuse angle with the 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 which 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 the 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

Where, s 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 right angle ,that is it is equal to 90°, the second angle is straight angle = 180°. As we know that 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.

FAQ (Frequently Asked Questions)

1. What is the Formula of the Right Angle Triangle?

Area of Right Angle Triangle = ½ (Base × Perpendicular)

If one of the angles is 90° and the other two angles are equal to 450 each, then the triangle is called an Isosceles Right Angled Triangle, where the adjacent sides to 90° are equal in length to each other.

2. What is the Perimeter of the Right Angle?

Perimeter of a right angled triangle =30cms. where a,b are two sides and c is the hypotenuse of a triangle. Also, it is given c=13cms. If b=5 then a=17−5=12.

3. How Do You Find the Area?

Area is measured in square units such as square inches, square feet or square meters. To find the area of any given rectangle, we need to multiply the length by the width. The formula for this is: A = Length * Width where A is equal to the area, L is equal to the length, W is equal to the width, and * means multiply.

4. What is the Area and Perimeter of a Right Triangle?

The perimeter of a triangle is simply the sum of its three sides. ... In simple (sort of), the Pythagorean theorem says that the sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of its hypotenuse. Every right triangle has three sides and ofcourse a right angle.