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Data Handling - Circle Graph or Pie Chart

Grade 8ICSE

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

🔑Concepts

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A Pie Chart (or Circle Graph) is a circular statistical graphic divided into sectors to illustrate numerical proportions. The entire circle represents the whole (100%100\% or 360∘360^{\circ}), while each sector represents a specific part of that whole.

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Each sector's size is proportional to the quantity it represents. Visually, a sector that looks like a quarter-circle indicates 25%25\% of the total (90∘90^{\circ}), while a semi-circle represents 50%50\% of the total (180∘180^{\circ}).

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The Central Angle is the angle subtended by the arc of a sector at the center of the circle. The magnitude of the data determines the width of this angle; the larger the data value, the wider the central angle.

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The sum of all central angles in any pie chart is always exactly 360∘360^{\circ}. This represents the total frequency or the total value of all components in the dataset.

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Data components can be expressed as fractions, percentages, or raw values. Before drawing a pie chart, these must be converted into degrees using the total sum of observations as the denominator.

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Visual interpretation allows for quick comparison. By looking at the relative areas of sectors, one can identify the 'mode' (the sector with the largest area) or compare the ratios between different categories without looking at the raw numbers.

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To construct a pie chart, you start by drawing a circle of a convenient radius and a horizontal radius. Each subsequent sector is drawn using a protractor, starting from the radius of the previous sector.

📐Formulae

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

Central Angle (from Percentage)=(Percentage Value100)×360∘\text{Central Angle (from Percentage)} = \left( \frac{\text{Percentage Value}}{100} \right) \times 360^{\circ}

Value of a Component=Central Angle360∘×Total Value\text{Value of a Component} = \frac{\text{Central Angle}}{360^{\circ}} \times \text{Total Value}

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

💡Examples

Problem 1:

In a survey of 7272 students, 1818 chose Blue as their favorite color, 2424 chose Red, and 3030 chose Green. Calculate the central angles for each color to represent this data on a pie chart.

Solution:

  1. Calculate the total number of students: 18+24+30=7218 + 24 + 30 = 72.
  2. Find the Central Angle for Blue: 1872×360∘=14×360∘=90∘\frac{18}{72} \times 360^{\circ} = \frac{1}{4} \times 360^{\circ} = 90^{\circ}.
  3. Find the Central Angle for Red: 2472×360∘=13×360∘=120∘\frac{24}{72} \times 360^{\circ} = \frac{1}{3} \times 360^{\circ} = 120^{\circ}.
  4. Find the Central Angle for Green: 3072×360∘=512×360∘=5×30∘=150∘\frac{30}{72} \times 360^{\circ} = \frac{5}{12} \times 360^{\circ} = 5 \times 30^{\circ} = 150^{\circ}.
  5. Verify the sum: 90∘+120∘+150∘=360∘90^{\circ} + 120^{\circ} + 150^{\circ} = 360^{\circ}.

Explanation:

To find the central angle, we divide the frequency of each color by the total frequency (7272) and multiply by 360∘360^{\circ}. The sum must equal 360∘360^{\circ} to ensure the entire circle is covered.

Problem 2:

A pie chart representing a family's budget shows a central angle of 108∘108^{\circ} for 'Food'. If the total monthly income is ₹45,000₹ 45,000, calculate the amount spent on food and the percentage of income it represents.

Solution:

  1. Calculate the amount spent on food: Amount=108∘360∘×45,000\text{Amount} = \frac{108^{\circ}}{360^{\circ}} \times 45,000.
  2. Simplify the fraction: 108360=310\frac{108}{360} = \frac{3}{10}.
  3. Final amount: 310×45,000=3×4,500=₹13,500\frac{3}{10} \times 45,000 = 3 \times 4,500 = ₹ 13,500.
  4. Calculate the percentage: Percentage=108∘360∘×100=310×100=30%\text{Percentage} = \frac{108^{\circ}}{360^{\circ}} \times 100 = \frac{3}{10} \times 100 = 30\%.

Explanation:

By knowing the central angle, we can determine the portion of the total. We multiply the ratio of the angle to 360∘360^{\circ} by the total money value to find the expense, and by 100100 to find the percentage.