Pie Charts
1. Core Idea
A pie chart shows parts of a whole. Every slice is a percentage of the total, and all slices sum to 100% (or 360 degrees).
The crucial limitation to teach: a pie chart shows proportions, not amounts. A 40% slice of a small pie can be far less than a 20% slice of a large one. Most GRE pie-chart traps are built on exactly this.
2. Must-Know Rules
| Concept | Formula |
|---|---|
| Slice value | percent x total |
| Percent from a slice | slice/total x 100 |
| Degrees from percent | percent x 3.6 |
| Percent from degrees | degrees / 3.6 |
| All slices sum to | 100% or 360 degrees |
The degree conversion: 1% = 3.6 degrees, and 10% = 36 degrees. A quarter of the pie is 90 degrees.
| Percent | Degrees |
|---|---|
| 10% | 36 |
| 20% | 72 |
| 25% | 90 |
| 33.3% | 120 |
| 50% | 180 |
Comparing two pie charts. If two pies represent different totals, percentages are not comparable directly. Convert both to absolute values first.
Example: Company A (total $50m) spends 30% on R&D = $15m. Company B (total $120m) spends 20% on R&D = $24m. B spends more despite the lower percentage.
Finding a missing slice: subtract the known slices from 100%.
3. Worked Examples
Example 1 — Slice to value
A budget of $180,000 is divided: Staff 45%, Equipment 20%, Rent 25%, Other 10%. How much goes to equipment?
0.20 x 180,000 = $36,000.
Example 2 — Degrees
A slice represents 15% of the total. What is its central angle?
15 x 3.6 = 54 degrees.
Example 3 — Two pies, different totals
School X (1,200 students) has 35% in Science. School Y (800 students) has 45% in Science. Which has more science students?
X: 0.35 x 1200 = 420.
Y: 0.45 x 800 = 360.
School X has more, despite the smaller percentage.
4. GRE Traps
- Comparing percentages across pies with different totals. The headline trap.
- Treating a pie slice as an absolute number. It is a share until you multiply by the total.
- Percent of a slice. "What percent of the Staff budget goes to training" uses the Staff amount as the base, not the whole budget.
- Forgetting slices must sum to 100%. If they don't, you've misread one.
- Degrees and percent confused. 25% is 90 degrees, not 25 degrees.
- Assuming visual slice size is exact. Read the labels.
- Percent change between two pies — you must convert both to values first.
5. Speed Tricks
- Convert every percent to a value once, at the start of the set, and reuse across all three questions.
- 1% = 3.6 degrees. Memorise it; degree questions then take five seconds.
- The missing slice = 100 minus the rest.
- For "which is larger" across two pies, always multiply out. Never compare the percentages.
- Halves and quarters by eye: a slice bigger than a quarter of the circle exceeds 25% — useful for eliminating choices.
- Watch for "of the X category" wording — that changes the base.
6. Self-Check
Q1. A slice is 40% of a total of 2,500. What is its value?
Q2. A slice has a central angle of 108 degrees. What percent of the total is it?
Q3. Pie A (total 400) has a 60% slice; Pie B (total 900) has a 30% slice. Which slice is bigger in absolute terms?
Answers
A1. 0.40 x 2,500 = 1,000.
A2. 108/3.6 = 30%.
A3. A: 240. B: 270. Pie B's slice is bigger.
7. One-Line Summary for the Board
A pie shows shares, not amounts. Multiply by the total before comparing across two pies. 1% = 3.6 degrees.