Pie charts

A pie chart is a circular picture divided into wedge-shaped slices, where each slice's area (or, equivalently, its central angle) is proportional to the share of the whole it represents. They are great for quickly comparing parts of a fixed total , like how a household spends its money or how many students like each subject.

Concept

Slice angle. A full circle is $360^\circ$. If a category accounts for the fraction $\dfrac{p}{T}$ of the total $T$, its slice spans

$\text{angle} = \dfrac{p}{T} \times 360^\circ.$

Example: a class of $40$ students has $10$ who chose football. The football slice spans $\dfrac{10}{40} \times 360 = 90^\circ$ , a quarter of the pie.

Steps to draw a pie chart.

1. List each category and its value.
2. Sum the values to get the total.
3. For each category, compute its slice angle.
4. Draw a circle. Mark a starting radius (often pointing up).
5. Use a protractor to mark off each slice angle in turn.
6. Label each slice with its category name and either its percent or its raw value.

Steps to read a pie chart.

1. Identify the total (sometimes given in the title, sometimes you must work it out from the labels).
2. To find the value of a slice, measure its angle and compute $\dfrac{\text{angle}}{360^\circ} \times \text{total}$.
3. To find what percent it represents, divide by $360^\circ$ and multiply by $100$.

When pie charts work well.

• Few categories ($2$-$6$).
• A clear "whole" that everything sums to.
• Comparing proportions, not absolute values.

When pie charts work poorly.

• More than $\sim 8$ slices (too many to compare).
• Categories with similar sizes (eye can't distinguish $30\%$ vs $32\%$ at a glance).
• Tracking change over time (a bar chart is better).
• When negative numbers or open-ended values are involved.

Worked examples

Example 1. A family of $40{,}000$ rupees per month spends: rent $\mathbb{?}\,12{,}000$, food $\mathbb{?}\,10{,}000$, transport $\mathbb{?}\,5{,}000$, savings $\mathbb{?}\,8{,}000$, others $\mathbb{?}\,5{,}000$. Find the slice angles.

Sum: $108 + 90 + 45 + 72 + 45 = 360^\circ$. ✓

Example 2. A pie chart shows that the slice for cricket has a central angle of $144^\circ$. If $400$ students were surveyed, how many like cricket?

• Number $= \dfrac{144}{360} \times 400 = \dfrac{2}{5} \times 400 = 160$.

Example 3. In a pie chart of $720$ items, three slices have angles $90^\circ, 120^\circ, 60^\circ$. Find the value of each.

• Slice 1: $\dfrac{90}{360} \times 720 = 180$.
• Slice 2: $\dfrac{120}{360} \times 720 = 240$.
• Slice 3: $\dfrac{60}{360} \times 720 = 120$.

Example 4. A pie chart shows percentages $30\%, 25\%, 20\%, 15\%, 10\%$. Find the angles.

• $30\% \to 108^\circ$, $25\% \to 90^\circ$, $20\% \to 72^\circ$, $15\% \to 54^\circ$, $10\% \to 36^\circ$.
• Sum: $360^\circ$. ✓

Try it yourself

1. A pie chart of $360$ children's favourite sport: cricket $144$, football $108$, hockey $72$, others $36$. Find the central angles.
2. Out of $500$ students, $200$ play cricket. Slice angle for cricket?
3. A pie chart shows angles $108^\circ$ for "rent" of a $\mathbb{?}\,30{,}000$ income. Find the rent amount.
4. Time spent by a student per day: study $4$ h, sleep $8$ h, school $6$ h, play $3$ h, others $3$ h. Compute slice angles.
5. A pie chart's slices in percent are $40\%, 25\%, 20\%, 10\%, 5\%$. Find their angles.
6. A budget of $\mathbb{?}\,80{,}000$ allocates $\mathbb{?}\,16{,}000$ to entertainment. Slice angle?
7. Why does the pie's slice angles always add to $360^\circ$?
8. Suppose a pie has $5$ equal slices. What is each angle?

Activity

My day in slices. Track how you spend $24$ hours one day. Round each activity to the nearest hour. Compute the slice angle for each and draw the pie. The slice for "school", "sleep", "phone", "play" will reveal your time priorities in one striking picture.