Bar charts, pictograms, pie charts — drawing and reading: Worked examples

Before we start, three plain-word reminders (every symbol earned before use):

The scenario matrix

Every question this topic can ask lives in one of these cells. The last column names the worked example that kills it.

#	Case class	What makes it tricky	Killed by
A	Bar chart — draw from data	convert value → cm via scale	Example 1
B	Bar chart — read a fractional bar	half-centimetre heights	Example 2
C	Pictogram — partial icon	reading 0.n of a symbol	Example 3
D	Pie chart — draw, awkward angles	fraction doesn't give a whole degree	Example 4
E	Pie chart — rounding fails the $36 0^{\circ}$ check	angles sum to $35 9^{\circ}$ / $36 1^{\circ}$	Example 5
F	Pie chart — reverse (angle → value)	undo the formula	Example 6
G	Zero / degenerate category	a value of $0$ , or one category = whole	Example 7
H	Real-world word problem (percentages first)	data given as $%$ , not counts	Example 8
I	Exam twist — mixed / missing value	back-out a hidden number	Example 9

We will hit every cell. Notice the maths tool used most is a single idea — the fraction of the whole — reused three ways. Ask yourself why a fraction and not, say, a subtraction? Because charts are about proportion: how big is one part relative to the whole. A fraction $\frac{v}{T}$ is exactly the machine that answers "what share?".

Cell A — Draw a bar chart

Cell B — Read a fractional bar

Cell C — Pictogram with a partial icon

Cell D — Pie chart with awkward angles

Worked example Example 4 (Cell D)

Data (favourite drink): Tea 7, Coffee 5, Juice 4. Draw the pie chart — find each sector angle.

Forecast: the total is 16, which does not divide 360 nicely — expect decimals.

Total $T = 7 + 5 + 4 = 16$ . Why this step? The whole circle ( $36 0^{\circ}$ ) must represent all the data, so we need the size of "all" first.
Tea angle $= \frac{7}{16} \times 36 0^{\circ} = 157. 5^{\circ}$ . Why the formula $\frac{v}{T} \times 360$ ? $\frac{v}{T}$ is Tea's share of the whole; multiplying that share by the whole circle turns the share into degrees. We use a fraction here because a pie chart is literally "parts of a whole".
Coffee angle $= \frac{5}{16} \times 36 0^{\circ} = 112. 5^{\circ}$ .
Juice angle $= \frac{4}{16} \times 36 0^{\circ} = 9 0^{\circ}$ (a clean right angle — a quarter of the circle).

The red Tea sector is the largest — it sweeps past the halfway line ( $18 0^{\circ}$ would be a straight edge; $157. 5^{\circ}$ is just short of it).

Verify: $157.5 + 112.5 + 90 = 36 0^{\circ}$ exactly ✓. The clean $9 0^{\circ}$ Juice sector is a nice sanity check: Juice is $\frac{4}{16} = \frac{1}{4}$ of the data, and $\frac{1}{4}$ of $360$ is $90$ .

Cell E — Rounding breaks the $36 0^{\circ}$ check

Worked example Example 5 (Cell E)

Data: A 3, B 3, C 3 (total 9). Each is $\frac{1}{3}$ of the pie, so each angle should be $12 0^{\circ}$ — but suppose a student rounds $\frac{3}{9} \times 360 = 119.99...$ to $120$ ... here we force the trap with $T = 7$ : A 2, B 2, C 3.

Forecast: $\frac{2}{7}, \frac{2}{7}, \frac{3}{7}$ won't give whole degrees; naive rounding may miss $360$ .

A $= \frac{2}{7} \times 360 = 102.857.. .^{\circ} \approx 10 3^{\circ}$ (rounded).
B $= \frac{2}{7} \times 360 \approx 10 3^{\circ}$ .
C $= \frac{3}{7} \times 360 = 154.285.. .^{\circ} \approx 15 4^{\circ}$ .
Check the sum: $103 + 103 + 154 = 36 0^{\circ}$ ? $103 + 103 + 154 = 360$ . ✓ — this one survives. Why check? Rounding each piece can nudge the total to $35 9^{\circ}$ or $36 1^{\circ}$ . Here it landed on $360$ by luck.

To show the fix when it fails, round differently: $103, 103, 155 = 36 1^{\circ}$ . Fix: keep the biggest sector (C) as the "adjuster": set $C = 360 - 103 - 103 = 15 4^{\circ}$ so the total is forced to $360$ .

Verify: exact values sum to $\frac{2}{7} \cdot 360 + \frac{2}{7} \cdot 360 + \frac{3}{7} \cdot 360 = \frac{7}{7} \cdot 360 = 36 0^{\circ}$ . Any rounded set must be nudged back to this exact $360$ .

Cell F — Reverse: angle back to a value

Cell G — Zero and degenerate categories

Worked example Example 7 (Cell G)

Data: Red 20, Blue 0, Green 20 in a pie chart. What angles?

Forecast: Blue has nothing — its slice should vanish. Red and Green split the rest.

Total $T = 20 + 0 + 20 = 40$ .
Blue $= \frac{0}{40} \times 360 = 0^{\circ}$ . Why? A share of $0$ is $0$ of anything. A zero category draws no slice at all — you simply omit it (do not leave a gap).
Red $= \frac{20}{40} \times 360 = 18 0^{\circ}$ , Green $= \frac{20}{40} \times 360 = 18 0^{\circ}$ . Why? Each is exactly half the data, so each takes half the circle — the pie becomes two equal semicircles.

The red Red-sector fills the top half; the zero Blue slice is invisible — the figure has only two edges, not three.

Degenerate extreme: if one category held all the data (say Red 40, others 0), its angle is $\frac{40}{40} \times 360 = 36 0^{\circ}$ — a full circle with no divisions. A pie chart of one category is just a disc.

Verify: $0 + 180 + 180 = 36 0^{\circ}$ ✓. And the all-Red case: $\frac{40}{40} \times 360 = 36 0^{\circ}$ ✓.

Cell H — Word problem given in percentages

Cell I — Exam twist: a missing value

Recall Quick self-test (reveal after trying)

A pie of 60 people: a sector is $7 2^{\circ}$ . How many people? ::: $\frac{72}{360} \times 60 = 12$ people. Pictogram key ⚽ = 8; you see $2.5$ icons. How many goals? ::: $2.5 \times 8 = 20$ goals. Bar scale 1 cm = 3 pets; bar is $4$ cm. How many pets? ::: $4 \times 3 = 12$ pets. Three sectors round to $103, 103, 155$ . What's wrong and the fix? ::: They sum to $36 1^{\circ}$ ; adjust the largest down to $15 4^{\circ}$ so the total is $36 0^{\circ}$ .

Connections

Parent topic (Hinglish) — the chart machines this page stress-tests
Fractions and percentages — every "share" step (Cells D, F, H) is a fraction/percent of the whole
Angles — measuring and summing the $36 0^{\circ}$ (Cells D, E, G, I)
Frequency tables — the raw values feeding each example
Ratio and proportion — scaling icons and bar heights
Averages and range — further statistics on the same datasets
Data collection methods — where survey data (Cell H) comes from