Understanding Pie Diagrams and Angle Calculation

A pie diagram, also known as a pie chart, represents data as slices of a circle. The entire circle represents the total data (100%) and has a central angle of 360°. Each slice's angle is proportional to the quantity it represents.

Initial Pie Diagram Analysis

The original pie diagram has four slices with the following central angles:

• Slice 1: 150°
• Slice 2: 90°
• Slice 3: 60°
• Slice 4: 60°

Let's check the total angle of the initial pie diagram:

\(150^\circ + 90^\circ + 60^\circ + 60^\circ = 360^\circ\)

This confirms that the initial angles form a complete pie diagram.

Forming the New Pie Diagram

A new pie diagram is formed by deleting one of the slices having an angle of 60°. This means one slice is removed from the original four.

The remaining slices have angles:

• Slice A: 150°
• Slice B: 90°
• Slice C: 60°

The sum of the angles of the remaining slices is:

\(150^\circ + 90^\circ + 60^\circ = 300^\circ\)

This sum (300°) represents the total of the remaining data. However, for the new pie diagram to be a complete circle, the total angle must be 360°. Therefore, the angles of the remaining slices must be rescaled proportionally so that their sum becomes 360°.

Calculating New Angles in the Rescaled Pie Diagram

The total angle of the remaining slices is 300°. This 300° needs to correspond to the full 360° of the new pie diagram. We can find the scaling factor by dividing the new total angle by the old total angle of the remaining slices.

Scaling Factor = \(\frac{\text{New Total Angle}}{\text{Old Total Angle of Remaining Slices}} = \frac{360^\circ}{300^\circ}\)

Scaling Factor = \(\frac{36}{30} = \frac{6}{5}\)

Now, we multiply each of the remaining slice angles by this scaling factor to find their new angles in the new pie diagram:

• New angle for the 150° slice: \(150^\circ \times \frac{6}{5} = (150 \div 5) \times 6 = 30 \times 6 = 180^\circ\)
• New angle for the 90° slice: \(90^\circ \times \frac{6}{5} = (90 \div 5) \times 6 = 18 \times 6 = 108^\circ\)
• New angle for the 60° slice: \(60^\circ \times \frac{6}{5} = (60 \div 5) \times 6 = 12 \times 6 = 72^\circ\)

The angles of the slices in the new pie diagram are 180°, 108°, and 72°.

Let's verify the sum of these new angles:

\(180^\circ + 108^\circ + 72^\circ = 360^\circ\)

The sum is 360°, which is correct for a complete pie diagram.

Identifying Largest and Smallest Slices in the New Diagram

The angles in the new pie diagram are 180°, 108°, and 72°.

• The largest angle is 180°.
• The smallest angle is 72°.

Evaluating the Options

Let's examine each option based on the angles calculated for the new pie diagram (180°, 108°, 72°).

• Option 1: The largest slice has angle 150°. Our calculated largest angle is 180°. So, this option is incorrect.
• Option 2: The smallest slice has angle 70°. Our calculated smallest angle is 72°. So, this option is incorrect.
• Option 3: The largest slice has angle 180°. Our calculated largest angle is 180°. So, this option is correct.
• Option 4: The smallest slice has angle 90°. Our calculated smallest angle is 72°. So, this option is incorrect.

Conclusion

Based on the calculations, the new pie diagram has slices with angles 180°, 108°, and 72°. The largest slice has an angle of 180°.

Revision Table: Pie Diagram Calculations

Additional Information: Working with Pie Charts and Proportionality

Pie charts are useful for visualizing how a whole is divided into parts. Each slice's size (both area and central angle) is directly proportional to the proportion of the total that the slice represents.

The proportion of a slice can be calculated as:

\(\text{Proportion} = \frac{\text{Slice Angle}}{360^\circ}\)

Or if you have the proportion (as a decimal or percentage), the angle is:

\(\text{Slice Angle} = \text{Proportion} \times 360^\circ\)

When you remove a slice and create a new pie diagram from the remaining ones, the remaining slices now represent 100% of the *new* total. The key is that the *relative* proportions of the remaining slices stay the same. If one remaining slice was twice as large (in angle) as another before rescaling, it will still be twice as large after rescaling.

For example, in the remaining slices (150°, 90°, 60°), the ratio is \(150:90:60\). Dividing by 30, this simplifies to \(5:3:2\). Let's check the new angles: 180°, 108°, 72°. Dividing by 36, this simplifies to \(5:3:2\). The ratios are indeed preserved.

The process of deleting a slice and forming a new diagram is equivalent to calculating the proportion of the *remaining total* that each remaining slice represents, and then multiplying that proportion by 360°.

Example for the 150° slice:

• Remaining total angle: 300°
• Proportion of 150° slice within the remaining total: \(\frac{150^\circ}{300^\circ} = \frac{1}{2}\)
• New angle in 360° chart: \(\frac{1}{2} \times 360^\circ = 180^\circ\)

This method gives the same results and helps understand the proportionality concept.