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Shapes and Angles - Degree Clock and Measuring Angles

Grade 5CBSE

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

🔑Concepts

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An angle is formed when two lines meet at a common point called the vertex. We measure the opening between these lines in degrees, denoted by the symbol ∘^\circ. A full circle or a complete turn is 360∘360^\circ.

A diagram showing two lines meeting at a vertex to form an angle.
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A Degree Clock uses the 12-hour clock face to help us understand angles. Since a full turn is 360∘360^\circ and there are 12 divisions, each hour mark represents a 30∘30^\circ jump (360∘÷12=30∘360^\circ \div 12 = 30^\circ). For example, at 2 o'clock, the angle is 2×30∘=60∘2 \times 30^\circ = 60^\circ.

A clock showing 2 o'clock where the angle between hands is 60 degrees.
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Angles are classified by their size: Acute angles are less than 90∘90^\circ (like an L-shape closing), Right angles are exactly 90∘90^\circ (like a perfect L-shape), and Obtuse angles are more than 90∘90^\circ but less than 180∘180^\circ (like an L-shape opening wider).

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We use a protractor to measure angles precisely. The center of the protractor is placed on the vertex, and the base line is aligned with one arm of the angle. We read the degree where the second arm crosses the scale.

A semi-circular protractor measuring a 45 degree angle.

📐Formulae

Right Angle=90∘\text{Right Angle} = 90^\circ

Straight Angle=180∘\text{Straight Angle} = 180^\circ

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

Acute Angle<90∘\text{Acute Angle} < 90^\circ

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

Angle=Fraction of Turn×360∘\text{Angle} = \text{Fraction of Turn} \times 360^\circ

💡Examples

Problem 1:

What is the measure of the angle formed by the hands of a clock at 3:00, and what type of angle is it?

Solution:

Step 1: A clock is divided into 1212 equal hour divisions. The total angle in a circle is 360∘360^\circ. Step 2: Calculate the angle for one hour division: 360∘12=30∘\frac{360^\circ}{12} = 30^\circ. Step 3: At 3:00, the minute hand is at 1212 and the hour hand is at 33. This is a gap of 33 hours. Step 4: Multiply the hours by the angle per hour: 3×30∘=90∘3 \times 30^\circ = 90^\circ.

Explanation:

Since the angle is exactly 90∘90^\circ, it forms a right angle, resembling the corner of a square.

Problem 2:

If a degree clock shows a 13\frac{1}{3} turn, calculate the angle in degrees and identify the type of angle.

Solution:

Step 1: A full turn in a degree clock is equal to 360∘360^\circ. Step 2: To find a 13\frac{1}{3} turn, multiply the fraction by the total degrees: 13×360∘\frac{1}{3} \times 360^\circ. Step 3: 360÷3=120∘360 \div 3 = 120^\circ.

Explanation:

The calculated angle is 120∘120^\circ. Since 120∘120^\circ is greater than 90∘90^\circ but less than 180∘180^\circ, it is classified as an obtuse angle.

Problem 3:

Calculate the angle between the hands of a clock when the time is exactly 4:00. What type of angle is this?

Clock showing 4 o'clock with an obtuse angle of 120 degrees between the hands.

Solution:

  1. Each hour division on a clock represents 360∘÷12=30∘360^\circ \div 12 = 30^\circ.
  2. At 4:00, the minute hand is at 12 and the hour hand is at 4.
  3. The number of hour divisions between 12 and 4 is 4.
  4. Angle = 4×30∘=120∘4 \times 30^\circ = 120^\circ.

Explanation:

Since the angle 120∘120^\circ is greater than 90∘90^\circ but less than 180∘180^\circ, it is an obtuse angle.

Problem 4:

A fan rotates through a 14\frac{1}{4} turn and then another 18\frac{1}{8} turn. Calculate the total angle covered in degrees.

A circle showing a sector representing a 135 degree turn.

Solution:

  1. We know a full turn is 360∘360^\circ.
  2. A 14\frac{1}{4} turn = 14×360∘=90∘\frac{1}{4} \times 360^\circ = 90^\circ.
  3. A 18\frac{1}{8} turn = 18×360∘=45∘\frac{1}{8} \times 360^\circ = 45^\circ.
  4. Total angle = 90∘+45∘=135∘90^\circ + 45^\circ = 135^\circ.

Explanation:

We calculate the degree measure for each fraction of a full circle (360∘360^\circ) and then add them together to find the total rotation.