Measuring Angles

An angle is a figure in plane geometry that is created by two rays or lines that have a shared endpoint. The Latin word "angulus," which means "corner," is where the English term "angle" originates. In real life, angles are used in many places such as carpenters use them for the precise building of chairs, and tables. Athletics use them to enhance their performance.

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In this article, we will cover the concept of the Measurement of angles. This category falls under the broader category of Trigonometry, which is a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination(JEE Main) and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Questions based on this topic have been asked frequently in JEE Mains.

What is an Angle?

An angle is a figure in plane geometry that is created by two rays or lines that have a shared endpoint. The Latin word "angulus," which means "corner," is where the English term "angle" originates.

An angle's common terminal is referred to as the vertex, and the two rays that make up its sides are called the side of an angle. The symbol "∠" is used to indicate the angle. The Greek letters θ, α, β, etc., can be used to indicate the angle measurement between the two rays.

If the angles are measured from the line, it is categorized into two parts :

1. Positive Angle
2. Negative angle

Positive angle -If the angle is measured in an anticlockwise direction it is called a Positive angle.

Negative angle - If the angle is measured in a clockwise direction it is called a negative angle.

Measurement of Angle

To form an angle, we start with two rays lying on top of one another. We leave one fixed in place and rotate the other. The fixed ray is the initial side, and the rotated ray is the terminal side. And, the measure of an angle is the amount of rotation from the initial side to the terminal side.

An angle is in standard position if its vertex is located at the origin, and its initial side extends along the positive x-axis, as you can see from the figure given below.

If the angle is measured in a counterclockwise direction from the initial side to the terminal side, the angle is said to be a positive angle. If the angle is measured in a clockwise direction, the angle is said to be a negative angle.

What is a Measuring Angle?

Measuring angles is done by using basic geometric tools like a protractor and a compass. These tools help in finding the exact measure of an angle. A protractor helps in providing the exact measure of the angle and a compass helps in constructing the angle. Measuring angles is done in three ways - degrees, radians, and revolution.

Degrees

The degree is the unit of measure of an angle and is measured by using the geometric tool - a protractor. A degree is denoted by the symbol '°'. A circle completely rotates at a 360° and a degree is a part of that 360° rotation as it is divided into 360 equal parts.

Radians

Radian is another unit of measurement of an angle and is used in place of degrees when the angle needs to be measured in terms of radians. By definition, a radian is the ratio of the length of the arc that the angle subtends of a circle, divided by the length of the radius of the same circle. In other words, a radian is an angle subtended by the arc of the length of the radius of the same circle at the center and the ratio will give the radian measure of the angle. Radian is denoted as rad.

Revolution

The revolution also is the unit of 360° as an angle is basically a subdivision of a circle rather than the sum of a few degrees.

System used for the measurement of angles

There are three systems used for the measurement of angles

1. Sexagesimal System

2. Centesimal system

3. Circular system

1. Sexagesimal system

In this system, an angle is measured in degrees, minutes, and seconds.

1 Right angle = 90^o (Read as 90 degrees)

1^o = 60’ ( 1 degree = 60 minutes)

1’ = 60” ( 1 minutes = 60 seconds)

2. Centesimal system

In this system, the measurement of the right angle is divided into 100 equal parts, and parts called Grades.

1 Right angle = 100g (Read as 100 grades)

3. Circular system

In this system, an angle is measured in radians.

One radian is the measure of the central angle of a circle that intercepts an arc equal in length to the radius of that circle. A central angle is an angle formed at the centre of a circle by two radii.

The formula for the radian measure of an angle formed by an arc of length l at the centre of the circle of radius r is (Length of arc)/(Radius) = l/r

Because the total circumference equals 2π times the radius, a full circular rotation is 2π radians.

So, 2π radians = 360°

So, π radians = 360°/ 2 = 180°

and 1 radian = 180°/ π ≈ 57.3°

Interconversion of units

1. Degree to Radian

Since, degrees and radians both measure angles, we need to be able to convert between them. We can easily do so using a proportion (where θ is the measure of the angle in degrees and θ_R is the measure of the angle in radians)

$\frac{\theta }{180}=\frac{{\theta }_{\mathrm{R}}}{\pi }$

Or

$\frac{\text{Degree}}{180}=\frac{\text{Radians}}{\pi }$

Note:

(i) Radian is the unit to measure angles, and it does not mean that π stands for 180^o. π is a real number. Remember the relation, π radians = 180^o.

(ii) In a circle of radius r, the length of an arc s is subtended by an angle with measure θ in radians. Arc length = (radius) x (Angle subtended by an arc in radians)

$\mathrm{s}=\mathrm{r}\theta$

2. Degree to grades

If we denote the number of degrees by D and the number of grades by G, the relation between them is given by

$\frac{D}{90}=\frac{G}{100}$

3. Radian to Grades

If the number of radians is represented by $R$ and the number of grades is represented by G, the relation between Radian and Grades is given by

$\frac{G}{100}=\frac{2\ast R}{\pi }$
Measurement of angles using Protractor

An angle is measured by using two geometric tools - a protractor and a compass. While a protractor can be used for both constructing and measuring, a compass is mostly used for constructing an angle. A protractor is considered one of the most important geometric tools as it helps in measuring angles in both degrees and radians

The steps to measure an angle are:

Step 1: Place the centre of the protractor on the vertex of the angle.

Step 2: Superimpose one side of the angle with the zero line of the protractor.

Step 3: The angle is equal to the number of degrees crossed on the protractor.

Constructing Angles Using a Protractor

A protractor can be used not only for measuring but also for constructing angles. This helps in both measuring the angles accurately and learning how to use the protractor.

The steps to construct an Angle:

Step 1: Draw a baseline.

Step 2: Mark the point O and place the centre of the protractor at O.

Step 3: Align the baseline of the protractor with the line.

Step 4: In the inner readings, look for the angle to be constructed and mark it as point C.

Step 5: Now using a scale, join O and C.

Summary

Measuring angles is a basic concept of geometry and trigonometry. It is essential for understanding relationships and solving a wide range of mathematical and practical problems. Accurate angle measurement is necessary for precise calculations. Whether using traditional instruments or digital tools, angle measurement enhances our ability to navigate, and design.

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Solved Examples Based on Measurement of Angles

Example 1: What is the shortest positive measure of the angle (in degrees) formed between the positive x-axis and the line if the total angle elapsed is ${810}^{\circ }$?

Solution: Total angle elapsed $={810}^{\circ }$
Total angle elapsed in two revolutions $={720}^{\circ }$
Thus, the angle $={810}^{\circ }-{720}^{\circ }={90}^{\circ }$
Hence, the answer is 90.

Example 2: What is the value of the radius of a circle if the circumference is $\frac{8\pi }{3}$?
Solution: Since the formula for circumference $=2\pi r$
Thus, $r=\frac{4}{3}$
Hence the answer is $\frac{-4}{3}$

Example 3: The angles of a triangle are in the ratio $2:3:5$. Find the least/minimum angle in radians.

Solution: Let the angles be $2\mathrm{x},3\mathrm{x},5\mathrm{x}$.
Thus sum $=10\mathrm{x}=1800$

$\Rightarrow x=180=18\times \frac{\pi }{180}=\frac{\pi }{10}\mathrm{rad}$
Minimum angle $=2x=2\times \frac{\pi }{10}=\frac{\pi }{5}$ radians.
Hence the answer is $\frac{\pi }{5}$ radian.

Example 4: If the radius of the circle = $\frac{1}{2}x$ circumference subtended by an angle A . Find the measure of angle A.

Solution: Circumference subtended by an angle = arc length

Also given that

r = l/2

So, l/r = 2

A = 2 radian (as l/r = angle in radians)

Hence the answer is 2 radian

Example 5: A circular wire of radius 3 cm is cut and bent so as to lie along a circle of radius 48 cm. Find the angle subtended by the wire at the centre of the circle.