Pie charts

A pie chart represents a whole as one full circle. The entire quantity being studied — all $40$ students in a class, the whole monthly income, the full $100\%$ of a data set — is represented by the complete circle, which is $360°$ of angle. Every category inside that whole gets a sector (a pie-slice), and the size of the sector's angle is set to match that category's proportion of the whole.

The sector-angle formula does the conversion. If a category's value is "part" out of a total "whole,"

where: part is the category's own value, whole is the true grand total of all categories combined, and $360°$ is the full angle of a circle. The fraction $\dfrac{\text{part}}{\text{whole}}$ is just the category's share; multiplying that share by $360°$ converts a share of the whole into a share of a circle's turning.

Percentages use the same idea with whole $=100\%$. When data is already given as percentages rather than raw counts,

so a category that is $25\%$ of the data gets a sector of $\dfrac{25}{100}\times360° = 90°$, a quarter of the circle.

Two checks guard every pie chart. First, the parts must add up to the whole — every category's value summed together must equal the stated total (or every percentage must sum to $100\%$). Second, the sector angles must add up to $360°$, which is really the same proportion applied to the whole itself: $\dfrac{\text{whole}}{\text{whole}} \times 360° = 360°$. If either check fails, there is an arithmetic slip somewhere before the chart is drawn.

Reading a chart runs the formula backwards. Given a sector's angle and the known whole, the actual value of that category is recovered with $\text{part} = \dfrac{\text{Sector angle}}{360°} \times \text{whole}$ — the same proportion, solved for the unknown part instead of the unknown angle.

Constructing a pie chart is a short, mechanical procedure once the angles are known. Calculate every sector angle first, using the same whole for every category. Draw a circle and mark its centre, then draw one radius as a fixed starting line — this is where the protractor's zero mark will sit. Working around the circle from that starting line, use the protractor to mark off each sector angle in turn, drawing a new radius at each mark, and label each sector as it is closed off. Because every angle was computed from the same whole, the last sector should close the circle exactly back at the starting line, which is the practical, hands-on version of the sum-to-$360°$ check.

In a class of $40$ students, favourite sports are: Cricket — $16$, Football — $10$, Badminton — $8$, Others — $6$. Find the sector angle for each sport, and use the result to also find how many students play Cricket if a second class of $60$ students shows the same $144°$ Cricket sector.

Step 1. Confirm the total: $16 + 10 + 8 + 6 = 40$, which matches the class size, so the parts-sum-to-whole check passes before any angle is calculated.

Step 2. Apply the sector-angle formula to each category using whole $=40$:

Step 3. Check the sum: $144° + 90° + 72° + 54° = 360°$, confirming no arithmetic slip, so the pie chart can be drawn with sectors of $144°$, $90°$, $72°$ and $54°$.

Step 4. For the second class, reverse the formula: $\text{part} = \dfrac{\text{Sector angle}}{360°} \times \text{whole} = \dfrac{144}{360} \times 60 = \dfrac{2}{5} \times 60 = 24$ students play Cricket.

The sector angles are $144°$, $90°$, $72°$, $54°$, and in the second class, $\mathbf{24}$ students play Cricket.

Competitive stretch

A family's monthly budget pie chart shows Rent $120°$, Food $90°$, Savings $60°$, Other expenses $90°$, with total income ₹60,000. Confirm the angles are valid first: $120° + 90° + 60° + 90° = 360°$, so the chart is internally consistent before any value is calculated. Reversing the formula for each category then gives Rent $= \dfrac{120}{360}\times60000 = ₹20{,}000$, Food $= \dfrac{90}{360}\times60000 = ₹15{,}000$, Savings $= \dfrac{60}{360}\times60000 = ₹10{,}000$, Other $= \dfrac{90}{360}\times60000 = ₹15{,}000$, and these sum to $₹60{,}000$, matching the given income exactly — a check that works precisely because Step 3 of the derivation above guarantees it.

• Forgetting the sum-to-$360°$ check. If computed sector angles do not add to $360°$ (allowing for small rounding), there is an arithmetic error to find before drawing anything.
• Using the wrong "whole." The whole must be the true grand total (all students, all income) — never one of the individual categories mistaken for the total.
• Rounding too early. Round each sector angle only at the final step; rounding intermediate fractions can push the total a few degrees away from $360°$.
• Confusing percentage with sector angle. A category that is $25\%$ of the whole has a sector angle of $90°$, not $25°$ — always convert through $\dfrac{\%}{100}\times360°$ before drawing.
• Reading slice size as the actual value. A bigger sector only means a bigger share; the real quantity also depends on what the whole represents, so recovering it needs the reverse formula, not a guess.
• Mixing up degrees between charts with different wholes. A $90°$ sector in a chart of $40$ students and a $90°$ sector in a chart of $60{,}000$ rupees represent the same fraction ($25\%$) but very different actual quantities — always re-apply the reverse formula with the correct whole for that specific chart rather than comparing angles across two unrelated pie charts.

Ask the tutor: Want to try building a pie chart from your own data set, step by step with the sector-angle formula and a check on the 360-degree sum?