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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 when two rays originate from a common endpoint called the vertex. Visually, imagine two straight lines meeting at a point; the amount of 'turn' or 'opening' between these two lines is the measure of the angle.

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The standard unit for measuring angles is degrees, represented by the symbol ∘^\circ. A complete rotation or a full circle represents 360∘360^\circ. If you imagine a clock, a full circle made by the minute hand is 360∘360^\circ.

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Angles are measured using a geometric tool called a protractor. It is a semi-circular device marked from 0∘0^\circ to 180∘180^\circ. It features two scales: an inner scale starting from 0∘0^\circ on the right and an outer scale starting from 0∘0^\circ on the left. To measure, the center point of the protractor must be placed exactly on the vertex of the angle.

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An Acute Angle is an angle that measures more than 0∘0^\circ but less than 90∘90^\circ. Visually, it looks 'sharp' and is narrower than a square corner.

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A Right Angle measures exactly 90∘90^\circ. It corresponds to 14\frac{1}{4} of a full revolution. Visually, it looks like the corner of a perfectly square book or the letter 'L'.

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An Obtuse Angle is greater than 90∘90^\circ but less than 180∘180^\circ. Visually, it appears wider than a right angle but is not yet a straight line.

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A Straight Angle measures exactly 180∘180^\circ and represents 12\frac{1}{2} of a full revolution. Visually, the two rays point in opposite directions, forming a single straight line.

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A Reflex Angle is an angle that is greater than 180∘180^\circ but less than 360∘360^\circ. It represents the 'outer' opening of an angle that is larger than a straight line.

📐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.