A pie chart is one of the friendlier-looking graphs: a circle split into coloured slices, with none of the axes and scales that put children off bar charts and line graphs. Yet the questions built on them catch a lot of pupils out — and a fair few parents helping with homework too. They come up again and again on 11+ papers, so they are well worth getting straight.

 

The reassuring part is that almost every pie chart question rests on a single idea: the whole circle stands for the whole group, and each slice is that group's share, written as an angle. Once that idea is secure, the rest is careful arithmetic.

 

Here is a typical example (adapted from a King's College School 11+ specimen paper):

A pie chart shows the favourite colour of a group of boys. The blue slice measures 76°, red is 84°, green is 62°, and the yellow slice is left as x°. We are asked to find x, then — given that red is the favourite of 126 boys — to work out how many boys chose each of the other colours, and finally the total number of boys. No calculator allowed, which matters more than it first appears.

 

THE ONE FACT EVERYTHING RESTS ON

A full circle is 360°, and on a pie chart those 360° represent the whole group — every boy sits somewhere in the circle. So each slice, measured in degrees, is in exact proportion to the number of boys it represents. A slice twice as large means twice as many boys. That is the whole trick, and every part of the question is just an application of it.

 

PART (a): THE MISSING ANGLE

Since the four slices together make up the complete circle, their angles must add up to 360°. Three are given, so the fourth is whatever is left over:

x = 360 − 76 − 84 − 62 = 138

The yellow slice is 138°. It is worth a quick check that the four angles now total 360: 76 + 84 + 62 + 138 = 360. They do.

 

PART (b): TURNING DEGREES INTO BOYS

This is the step that causes the trouble, so it is worth slowing right down.

We are told that red — an 84° slice — is the favourite of 126 boys. That one fact ties degrees to boys, and from it we can find what a single degree is worth:

84° represents 126 boys so 1° represents 126 ÷ 84 boys

With no calculator to hand, we simplify the fraction 126/84 rather than attempt the division directly, cancelling a little at a time:

126/84 = 63/42 = 9/6 = 3/2

The middle step, 63/42 down to 9/6, means spotting that 7 divides into both — the seven times table turning up where it is least welcome is exactly the sort of thing these papers enjoy. We are left with:

1° = 3/2 boys = 1.5 boys

That figure — one and a half boys per degree — is the key that opens every other slice. We simply multiply each angle by 1.5:

(i) Green is 62°, so 62 × 1.5 = 93 boys.

(ii) Yellow is 138° (from part a), so 138 × 1.5 = 207 boys.

(iii) Not blue means everyone except those who chose blue. Add up the other three slices, in boys:

green + red + yellow = 93 + 126 + 207 = 426 boys

One word of warning on this last part, because it is a classic slip: red counts as 126 boys — the number we were given — not 84. The 84 is red's angle in degrees; the figure we add here is the number of boys. Reaching for 84 out of habit gives the wrong total, so it pays to be deliberate about which quantity each number is.

 

PART (c): THE TOTAL

The whole pie chart is 360°, and each degree is worth 1.5 boys, so:

total = 360 × 1.5 = 540 boys

You do not even need the earlier parts for this — but it is a good habit to check the total against everything else. The four colours should add back up to it: 114 (blue) + 126 (red) + 93 (green) + 207 (yellow) = 540. And as a bonus, this gives a second route to part b(iii): not blue = 540 − 114 = 426, agreeing with the figure we found by adding. When two different methods land on the same answer, you can be confident it is right.

 

THE METHOD IN THREE STEPS

Strip away the colours and every pie chart question follows the same path:

1. The angles add up to 360° — use that to find any missing angle.
2. Find what one degree is worth, from whatever fact links the chart to real numbers (here, 84° = 126 boys, so 1° = 1.5 boys).
3. Multiply each angle by that value to find what each slice represents.

Master those three steps and a pie chart holds no surprises, whatever the slices happen to be coloured — and whatever times table the examiner decides to hide in the fractions.