In a pi-diagram there are three sectors. If the ratio of the angles of the sectors is 1 : 2 : 3, then what is the angle of the largest sector?
180°
Calculating Pie Chart Sector Angles from Ratios
A pie diagram, also known as a pie chart, is a circular graph divided into sectors. Each sector represents a proportion of the whole. The total angle in a circle is 360 degrees.
In this question, we have a pie diagram with three sectors. The ratio of the angles of these sectors is given as 1 : 2 : 3. This means that if we represent the smallest angle by a variable, say \(x\), the other two angles will be \(2x\) and \(3x\).
The sum of the angles of all the sectors in a pie diagram must equal the total angle of the circle, which is 360 degrees.
So, we can write an equation:
\(\text{Sum of angles} = \text{Angle of sector 1} + \text{Angle of sector 2} + \text{Angle of sector 3}\)
\(360^\circ = x + 2x + 3x\)
Now, we need to solve this equation for \(x\) to find the value of one part of the ratio.
\(360^\circ = (1 + 2 + 3)x\)
\(360^\circ = 6x\)
To find \(x\), divide 360 by 6:
\(x = \frac{360^\circ}{6}\)
\(x = 60^\circ\)
Now that we know the value of \(x\), we can find the angle of each sector:
- Angle of the first sector (ratio 1 part): \(x = 60^\circ\)
- Angle of the second sector (ratio 2 parts): \(2x = 2 \times 60^\circ = 120^\circ\)
- Angle of the third sector (ratio 3 parts): \(3x = 3 \times 60^\circ = 180^\circ\)
The question asks for the angle of the largest sector. Comparing the three angles we calculated (\(60^\circ\), \(120^\circ\), and \(180^\circ\)), the largest angle is \(180^\circ\).
Let's verify that the sum of these angles is indeed \(360^\circ\):
\(60^\circ + 120^\circ + 180^\circ = 360^\circ\)
This confirms our calculations are correct.
Therefore, the angle of the largest sector in the pie diagram is \(180^\circ\).
Revision Table: Pie Chart Angles & Ratios
| Concept | Explanation | Formula/Example |
|---|---|---|
| Pie Diagram Total Angle | The total angle in a circle, representing the whole dataset. | \(360^\circ\) |
| Ratio of Angles | Proportional distribution of the 360 degrees among sectors. | e.g., \(a : b : c\) |
| Finding Sector Angle | If ratio is \(a:b:c\), sum of ratio parts \(= a+b+c\). Angle of sector corresponding to ratio part 'a' is \(\frac{a}{a+b+c} \times 360^\circ\). | For ratio 1:2:3, total parts = 6. Angle of largest sector (3 parts) = \(\frac{3}{6} \times 360^\circ = \frac{1}{2} \times 360^\circ = 180^\circ\). |
Additional Information on Data Representation with Pie Charts
Pie charts are useful for showing the proportion of each category relative to the whole. Each sector's area (and consequently its angle at the center) is proportional to the frequency or value it represents. They are best used when you have a relatively small number of categories (ideally 4-6) and want to show how each part contributes to the total.
- Advantages: Visually intuitive for showing proportions, easy to compare relative sizes of sectors.
- Disadvantages: Can be difficult to compare values between different pie charts, hard to compare sectors with similar sizes, not suitable for large numbers of categories.
- Calculation Basis: The central angle of each sector is calculated as \(\text{Angle} = \frac{\text{Value of category}}{\text{Total Value}} \times 360^\circ\). When ratios are given, the ratio parts act as the 'values' and the sum of the ratio parts acts as the 'total value'.
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