Visual walkthrough — Bar charts, pictograms, pie charts — drawing and reading

This page slows down the pie-chart formula from the parent note and re-derives it one picture at a time.

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Step 1 — What "the whole" looks like

WHAT. Before we cut anything, we agree on the "whole". Collect every category's value and add them up. Call that total $T$.

$T=\underset{\text{first category}}{\underbrace{v_1}}+\underset{\text{second}}{\underbrace{v_2}}+\cdots +\underset{\text{last category}}{\underbrace{v_n}}$

Here each $v$ (for value) is just a count — how many apples, how many students. The little numbers below ($1,2,\dots ,n$) are name-tags telling apart the categories. $T$ (for Total) is the size of the entire pile.

WHY. A slice only means something compared to the whole. "10 apples" tells you nothing about slice size until you know it's 10 out of what. $T$ is that "out of what".

PICTURE. The bars on the left are the raw pile. On the right they are melted together into one solid block of height $T$ — the whole we are about to cut into a circle.

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Step 2 — What "a share" means (the fraction)

WHAT. For one category with value $v$, form the number $f=\frac{\stackrel{\text{this category’s count}}{\overbrace{v}}}{\underset{\text{whole pile}}{\underbrace{T}}}$ We call $f$ the fraction (or share) of that category. It is always between $0$ and $1$.

WHY division and not subtraction? We want proportion, not leftover. "What part of the whole is this?" is a division question — see Fractions and percentages. Subtraction ($T-v$) would tell us what's left over, a different question.

PICTURE. The block of height $T$ from Step 1 is sliced horizontally. Bananas take up 15 of the 30 units — exactly half the block. That "half" is the fraction $f=\frac{15}{30}=\frac{1}{2}$.

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Step 3 — Why a circle is worth 360°

WHAT. We are going to draw the whole pile as one full circle. A full turn around a circle is defined to be $360°$ — see Angles. So the entire pile $T$ corresponds to $360°$.

WHY 360 and not 100 or 1? It is a convention (chosen long ago because $360$ divides evenly by so many numbers). We could use $100$ (percent) or $2\pi$ (radians) instead — the logic is identical: the whole equals one full turn, whatever number we call it. We pick $360°$ because protractors are marked that way.

PICTURE. One empty circle. A red arrow sweeps all the way round and lands back where it started — that complete sweep is labelled $360°$. This is our "whole", the same whole as the $T$-block, just bent into a ring.

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Step 4 — Stretching the fraction onto the circle

WHAT. A category owns fraction $f=\frac{v}{T}$ of the pile. It should own the same fraction of the circle. Fraction $f$ of $360°$ is: $\theta =\underset{\text{share, }0\text{ to }1}{\underbrace{f}}\times \underset{\text{the full turn}}{\underbrace{360°}}$ $\theta$ (Greek letter theta) is the slice's angle, measured in degrees at the centre.

WHY multiply? "Fraction of something" always means multiply. Half of 360 is $\frac{1}{2}\times 360$. A quarter of 360 is $\frac{1}{4}\times 360$. Multiplying by $f$ scales the full turn down to just this category's part.

PICTURE. The horizontal fraction-bar from Step 2 is bent around the centre of the circle. A share of $f=\frac{1}{2}$ swings the arrow halfway round — a $180°$ slice.

Now substitute $f=\frac{v}{T}$ back in and we have the whole formula:

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Step 5 — Cutting the actual slices (our numbers)

WHAT. Feed each value through $\theta =\frac{v}{T}\times 360°$ with $T=30$.

${\theta }_{\text{Apples}}=\frac{10}{30}\times 360°=\frac{1}{3}\times 360°=120°$ ${\theta }_{\text{Bananas}}=\frac{15}{30}\times 360°=\frac{1}{2}\times 360°=180°$ ${\theta }_{\text{Oranges}}=\frac{5}{30}\times 360°=\frac{1}{6}\times 360°=60°$

Read each line as: count over total gives the share $\left(\frac{1}{3},\frac{1}{2},\frac{1}{6}\right)$, then times 360 turns the share into degrees.

WHY these look right. Apples are one-third of the pile, so they eat one-third of the turn ($120°$). Bananas are half, so a straight-line slice ($180°$). Oranges one-sixth ($60°$).

PICTURE. The finished pie, sectors coloured and each angle marked at the centre. Start at 12 o'clock, sweep clockwise, laying $120°$ then $180°$ then $60°$.

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Step 6 — The safety check (angles must close the circle)

WHAT. Add every slice angle. It must equal $360°$. $120°+180°+60°=360°✓$

WHY it must. The shares $f$ add to $1$ (the parts make the whole), and $1\times 360°=360°$. Algebraically: $\sum {\theta }_i=\sum \frac{v_i}{T}\times 360°=\frac{\sum v_i}{T}\times 360°=\frac{T}{T}\times 360°=360°$ Each symbol: $\sum$ means "add them all"; $\sum v_i=T$ because summing every category is the total; $\frac{T}{T}=1$.

PICTURE. The three slices snapped side by side on a number line of degrees, filling exactly the $0°\to 360°$ span with no gap and no overhang.

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Step 7 — The edge and degenerate cases

Every scenario must be covered, or the reader hits a wall you never showed.

Case A — a category with $v=0$. $\theta =\frac{0}{T}\times 360°=0°$ A zero-value category gets a zero-width slice: it simply doesn't appear. That's correct — nothing was there.

Case B — one category is everything ($v=T$). $\theta =\frac{T}{T}\times 360°=360°$ The single slice is the whole circle — no cut lines at all. Also correct.

Case C — total is zero ($T=0$). Then $\frac{v}{T}=\frac{0}{0}$ is undefined — you cannot divide by zero. Meaning: with no data at all, there is no pile to slice, so a pie chart is impossible. Not a formula failure — an honest "there's nothing to draw".

PICTURE. Three mini-circles: an all-blue full circle ($v=T$), a plain circle with a faint "0° — invisible" tag ($v=0$), and a greyed-out circle with a "no data" cross ($T=0$).

Recall Quick self-test

A category has $v=0$ out of $T=40$. What is its angle? ::: $\frac{0}{40}\times 360°=0°$ — an invisible slice. Why can't you draw a pie chart when $T=0$? ::: You'd divide by zero; there is literally no data, nothing to cut.

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The one-picture summary

The whole derivation on a single canvas: raw bars → melted block $T$ → one slice's share $\frac{v}{T}$ → that share bent onto $360°$ → the finished, angle-labelled pie.

Recall Feynman retelling (plain words)

Imagine dumping all your counts into one heap; the size of the heap is $T$. Pick one flavour — say bananas, 15 out of 30. That's half the heap, and "half" is just $15$ divided by $30$. Now take a whole pizza. A whole pizza is one full turn, and a full turn is $360°$. Bananas own half the heap, so they own half the pizza — half of $360°$ is $180°$, a straight-line slice. Do that for every flavour: count over total gives its share, share times $360°$ gives its slice. Line up all the slices and they fill the circle exactly, because all the shares add to one whole. If a flavour has zero, its slice is zero degrees — you just don't see it. And if the whole heap is empty, there's simply no pizza to cut. That's the entire pie chart, start to finish.

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Connections

• Fractions and percentages — the share $\frac{v}{T}$ is a fraction; every step lives here
• Angles — why a full turn is $360°$ and how to lay a sector with a protractor
• Ratio and proportion — "fraction of the whole" is proportional reasoning
• Frequency tables — where the values $v_1,\dots ,v_n$ come from
• Hinglish version of the parent