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Understanding Elementary Shapes - Measuring Angles

Grade 6CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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An angle is formed by two rays sharing a common endpoint called the vertex. The measure of an angle is typically expressed in degrees (∘^{\circ}), where a full revolution is divided into 360360 equal parts.

Diagram showing an angle with rays and a vertex.
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Angles are classified by their measures: An acute angle is less than 90∘90^{\circ}, a right angle is exactly 90∘90^{\circ}, and an obtuse angle is greater than 90∘90^{\circ} but less than 180∘180^{\circ}.

A right angle measuring 90 degrees.
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A protractor is the standard tool used to measure angles. It has two scales (inner and outer) marked from 0∘0^{\circ} to 180∘180^{\circ}.

Illustration of a protractor scale.
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A reflex angle is an angle whose measure is greater than 180∘180^{\circ} but less than 360∘360^{\circ}. It represents more than a straight line but less than a full circle.

📐Formulae

Full Revolution=360∘\text{Full Revolution} = 360^\circ

Half Revolution (Straight Angle)=180∘\text{Half Revolution (Straight Angle)} = 180^\circ

One-fourth Revolution (Right Angle)=90∘\text{One-fourth Revolution (Right Angle)} = 90^\circ

Acute Angle:0∘<θ<90∘\text{Acute Angle}: 0^\circ < \theta < 90^\circ

Obtuse Angle:90∘<θ<180∘\text{Obtuse Angle}: 90^\circ < \theta < 180^\circ

Reflex Angle:180∘<θ<360∘\text{Reflex Angle}: 180^\circ < \theta < 360^\circ

Measure of angle in degrees=(Fraction of revolution)×360∘\text{Measure of angle in degrees} = (\text{Fraction of revolution}) \times 360^\circ

💡Examples

Problem 1:

What fraction of a clockwise revolution does the hour hand of a clock turn through, when it goes from 3 to 9? Also, find the measure of the angle in degrees.

Solution:

Step 1: Total positions on a clock face = 1212. \nStep 2: Number of positions moved from 3 to 9 = 9−3=69 - 3 = 6. \nStep 3: Fraction of revolution = 612=12\frac{6}{12} = \frac{1}{2}. \nStep 4: Angle in degrees = 12×360∘=180∘\frac{1}{2} \times 360^\circ = 180^\circ.

Explanation:

Moving from 3 to 9 covers exactly half the clock face, which represents a straight angle or half a revolution.

Problem 2:

Classify the following angles based on their measures: (a) 45∘45^\circ (b) 170∘170^\circ (c) 210∘210^\circ.

Solution:

Step 1: For 45∘45^\circ, since 0∘<45∘<90∘0^\circ < 45^\circ < 90^\circ, it is an Acute Angle. \nStep 2: For 170∘170^\circ, since 90∘<170∘<180∘90^\circ < 170^\circ < 180^\circ, it is an Obtuse Angle. \nStep 3: For 210∘210^\circ, since 180∘<210∘<360∘180^\circ < 210^\circ < 360^\circ, it is a Reflex Angle.

Explanation:

Classification is determined by comparing the given angle measure to the standard benchmarks of 90∘90^\circ and 180∘180^\circ.

Problem 3:

Identify the measure of the angle shown in the diagram and classify it. The angle represents a turn from the 1212 o'clock position to the 44 o'clock position on a circular clock face.

Clock showing the angle between 12 and 4.

Solution:

  1. A full clock revolution is 360∘360^{\circ} and is divided into 1212 hours.
  2. Each hour represents 360∘12=30∘\frac{360^{\circ}}{12} = 30^{\circ}.
  3. From 1212 to 44 is 44 hours.
  4. Measure =4×30∘=120∘= 4 \times 30^{\circ} = 120^{\circ}.
  5. Since 90∘<120∘<180∘90^{\circ} < 120^{\circ} < 180^{\circ}, it is an obtuse angle.

Explanation:

By calculating the angle between the clock hands using the property that each hour mark represents 30∘30^{\circ}, we find the total degrees and then compare it against angle classification rules.

Problem 4:

Determine the measure of the smaller angle between the north and north-east directions on a compass.

Compass showing North, East and North-East directions.

Solution:

  1. The four main directions (N, E, S, W) divide the circle into 44 equal parts of 90∘90^{\circ} each.
  2. North-East (NE) bisects the angle between North and East.
  3. Angle =90∘2=45∘= \frac{90^{\circ}}{2} = 45^{\circ}.
  4. Since 45∘<90∘45^{\circ} < 90^{\circ}, it is an acute angle.

Explanation:

Since the North and East directions are perpendicular (90∘90^{\circ}), the intermediate direction North-East exactly halves that angle.