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Data Handling - Circle Graph or Pie Chart

Grade 8CBSE

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

🔑Concepts

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A Circle Graph or Pie Chart is a visual representation used to show the relationship between a whole and its parts. The entire circle represents the total value (100%100\% or 360∘360^{\circ}), and it is divided into slices known as sectors.

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Each 'Sector' in the graph represents a specific category of data. Visually, a sector is a part of the circle enclosed by two radii and the intercepted arc. The area and size of the sector are directly proportional to the value of the data it represents.

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The 'Central Angle' is the angle formed at the center of the circle by the radii of a sector. Since the sum of all angles around the center of a circle is 360∘360^{\circ}, the sum of all central angles in a pie chart must always equal 360∘360^{\circ}.

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To determine the size of a sector, data is converted into a fraction of the whole. For example, if a category represents half of the total data, it will be shown as a semi-circle with a central angle of 180∘180^{\circ}.

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Pie charts are most effective for displaying relative proportions rather than exact amounts. By looking at the size of the sectors relative to each other, one can easily identify which category is the most or least popular.

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The process of constructing a pie chart involves calculating the central angle for each component using the formula based on 360∘360^{\circ}, and then drawing these angles starting from a common radius using a protractor.

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Data in a pie chart can also be represented as percentages. In such cases, the entire circle represents 100%100\%, and a sector representing 25%25\% of the data would visually appear as a quarter-circle or a right-angled sector (90∘90^{\circ}).

📐Formulae

Fraction of a Component=Value of the ComponentTotal Value\text{Fraction of a Component} = \frac{\text{Value of the Component}}{\text{Total Value}}

Central Angle=(Value of the ComponentTotal Value)×360∘\text{Central Angle} = \left( \frac{\text{Value of the Component}}{\text{Total Value}} \right) \times 360^{\circ}

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

Percentage of a Component=(Central Angle360∘)×100\text{Percentage of a Component} = \left( \frac{\text{Central Angle}}{360^{\circ}} \right) \times 100

💡Examples

Problem 1:

In a school of 720 students, 180 students like Mathematics, 240 like Science, and 300 like English. Calculate the central angle for each subject to represent this data in a pie chart.

Solution:

Step 1: Identify the total value. Total students = 180+240+300=720180 + 240 + 300 = 720.\nStep 2: Calculate the central angle for Mathematics: 180720×360∘=14×360∘=90∘\frac{180}{720} \times 360^{\circ} = \frac{1}{4} \times 360^{\circ} = 90^{\circ}.\nStep 3: Calculate the central angle for Science: 240720×360∘=13×360∘=120∘\frac{240}{720} \times 360^{\circ} = \frac{1}{3} \times 360^{\circ} = 120^{\circ}.\nStep 4: Calculate the central angle for English: 300720×360∘=512×360∘=150∘\frac{300}{720} \times 360^{\circ} = \frac{5}{12} \times 360^{\circ} = 150^{\circ}.\nVerification: 90∘+120∘+150∘=360∘90^{\circ} + 120^{\circ} + 150^{\circ} = 360^{\circ}.

Explanation:

We find the central angle by taking the value of each subject as a fraction of the total student count and multiplying by 360∘360^{\circ}.

Problem 2:

A pie chart showing the expenditure of a family has a central angle of 108∘108^{\circ} for 'Food'. If the total monthly income is 50,000, calculate the amount spent on food.

Solution:

Step 1: Use the relationship between the central angle and the total value.\nStep 2: Amount spent on Food = Central Angle360∘×Total Value\frac{\text{Central Angle}}{360^{\circ}} \times \text{Total Value}.\nStep 3: Substitute the values: Amount=108∘360∘×50,000\text{Amount} = \frac{108^{\circ}}{360^{\circ}} \times 50,000.\nStep 4: Simplify the fraction: 108360=310\frac{108}{360} = \frac{3}{10}.\nStep 5: Calculate the final amount: 310×50,000=15,000\frac{3}{10} \times 50,000 = 15,000.

Explanation:

To find the actual value from a pie chart, we divide the specific sector's angle by the total angle of the circle (360∘360^{\circ}) and multiply by the total quantity.