What is an Angle?
How much a line is turned about a point with respect to a fixed line is actually called an ANGLE.
Activity 1 – Exploring Angle Measurement Systems
A class is divided into two groups. The group on the left is called the Babylonian / British group and the group on the right is called the French group.
A circular piece of paper of certain radius is given to all students of Class 9.
Instructions to Students
- Fold the paper into half.
- Again fold it into half.
- Unfold the paper completely.
- Identify the centre point where the two crease lines intersect.
- Trace the two crease lines with a pen or pencil to clearly divide the circle into four equal quadrants.
- At the centre, the angle formed is a right angle.
➜ The Babylonian group divides/slices one of their right angles into 90 equal parts.
➜ The French group divides/slices one of their right angles into 100 equal parts.
Observation
- Babylonians observe that their full circle will have $90 \times 4 = \mathbf{360}$ parts.
- The French observe that their full circle will have $100 \times 4 = \mathbf{400}$ parts.
- The physical right angle formed on both groups is exactly the same.
- The actual size of a right angle always stays exactly the same.
- Like a piece of cloth is measured in inches or metres, a right angle can also be measured by slicing it into 90 or 100 pieces.
Systems of Measurement of Angles
Based on 90 or 100 pieces of a right angle, there are two systems of measurement of angles.
a) Sexagesimal System (Babylonian / British System)
In this system, one right angle is divided into 90 equal parts. Each part is called a degree (°). Further, each degree is divided into 60 equal parts — each part is called a minute. Finally, each minute is divided into 60 equal parts — each part is called a second.
b) Centesimal System (The French System)
In this system, one right angle is divided into 100 equal parts. Each part is called a grade (g). Further, each grade is divided into 100 equal parts — each part is called a minute. Finally, each minute is divided into 100 equal parts — each part is called a second.
Relation between Degree and Grade Measure
The actual size of a right angle always stays the same, whether measured as 90 parts or 100 parts. Based on this fact we can write:
Again, if we write,
Activity 2 – Discovering the Radian
Students of Class 9 are given circular pieces of paper of different radii. Along with the paper, a thread is also given.
Instructions to Students
- Measure the radius of the circular paper with the thread and cut the thread equal to the radius.
- Wrap the thread (equal to radius) around the circumference of the circular paper.
- Mark the ends of the thread on the circumference.
- Join the ends to the centre of the circular paper.
- An angle is formed at the centre.
Observation
The opening of the angle at the centre is the same even though the size of the circular paper is different for different students.
- Measurement of angle does not depend on the number of slices.
- The natural and universal method to measure angles — using the radius and the length along the circumference (arc length) — is the Radian System.
c) Radian System
In this system, the unit of measurement is Radian (c).
1c (One Radian)
One radian (1c) is the angle subtended (formed) at the centre of a circle by an arc equal to the radius of that circle.
Theorem 1 – A Radian is a Constant Angle
"A Radian is a constant angle."
Given
A circle with centre O and radius $r$.
Such that: $OA = OB = AB = r$ | $\angle AOB = 1^c$ (by definition) | $\angle AOC = 180°$ (straight angle)
| # | Statements | Reasons |
|---|---|---|
| 1. | $\angle AOB \stackrel{\circ}{=} \widehat{AB}$ $\angle AOC \stackrel{\circ}{=} \widehat{ABC}$ |
Relation between central angle and arc. |
| 2. | $\dfrac{\angle AOB}{\angle AOC} = \dfrac{\widehat{AB}}{\widehat{ABC}}$ | Dividing (1) by (2). |
| 3. |
$\dfrac{1^c}{180°} = \dfrac{r}{\pi r}$ $\dfrac{1^c}{180°} = \dfrac{1}{\pi}$ |
Substituting given values. |
| 4. | $\therefore\; 1^c = \dfrac{180°}{\pi}$ | $180°$ and $\pi^c$ are constant values. |
∴ A radian is constant.
Theorem 2 – Relation between Central Angle, Arc Length, and Radius
To establish the relation between Central Angle, Arc Length, and Radius.
Given
A circle with centre O and radius $r$.
Such that: $OA = OB = OC = AB = r$
$\angle AOB = 1^c$ (by definition)
$\angle AOC = \theta$ (say)
$\widehat{AB} = r$, $\widehat{AC} = l$ (say)
| # | Statements | Reasons |
|---|---|---|
| 1. | $\angle AOB \stackrel{\circ}{=} \widehat{AB}$ $\angle AOC \stackrel{\circ}{=} \widehat{AC}$ |
Relation between central angle and arc. |
| 2. | $\dfrac{\angle AOC}{\angle AOB} = \dfrac{\widehat{AC}}{\widehat{AB}}$ | Dividing the given relations. |
| 3. |
$\dfrac{\theta}{1^c} = \dfrac{l}{r}$ or $\theta = \dfrac{l}{r} \times 1^c$ |
Substituting given values. |
For a circle of radius $r$ with an arc of length $l$ subtending angle $\theta$ at centre:
Relation between Degree, Grade, and Radian
From Theorem 1: $\;1^c = \dfrac{180°}{\pi}$
We know: $\;180° = 200^g \;\cdots\!(2)$
$\Rightarrow 1^c \times \pi = 180°$
$\Rightarrow \pi^c = 180° \;\cdots\!(1)$
From (1) and (2):
Deriving Individual Relations
| From | Degree → Radian | Radian → Degree |
|---|---|---|
| $180° = \pi^c$ | $1° = \dfrac{\pi^c}{180}$ | $1^c = \left(\dfrac{180}{\pi}\right)°$ |
| $200^g = \pi^c$ | $1^g = \dfrac{\pi^c}{200}$ | $1^c = \left(\dfrac{200}{\pi}\right)^g$ |
| $180° = 200^g$ | $1° = \left(\dfrac{10}{9}\right)^g$ | $1^g = \left(\dfrac{9}{10}\right)°$ |
- Sum of all angles of a triangle in degree is $180° = 2 \times 90° = 2 \times 100^g = 200^g = \pi^c$
- Sum of all angles of a quadrilateral in degree is $360° = 4 \times 90° = 4 \times 100^g = 400^g = 2\pi^c$