To find the angle of a sector, divide its count by the total and multiply by 360°, so 9 out of 24 is 3/8 of a turn, or 135°. The angles of all the categories must add up to 360°, and that sum is your check. If you only need each category’s fraction, the what fraction is it calculator finds it and simplifies it. If you want a table of equivalences between fractions of a turn and degrees, it is in the guide on fractions of a turn, degrees and radians.
| You have | Angle calculation | Example |
|---|---|---|
| Counts | count ÷ total × 360° | 9 of 24 → 135° |
| Fractions | fraction × 360° | 1/8 → 45° |
| Percentages | percentage × 3.6° | 10% → 36° |
All three rows start from the fraction of the circle that the category takes up; only the way the data arrives changes. If you have the fraction and want it as a percentage, use the fraction to percent conversion. The rest of this guide uses the first row, the most common one in a survey.
The fraction of a number tool does step 3 with any fraction. Here the angles come out whole because 36 divides 360, and each student takes up 360 ÷ 36 = 10°.
| Way to school | Students | Fraction | Angle |
|---|---|---|---|
| Bus | 15 | 5/12 | 150° |
| Walk | 12 | 1/3 | 120° |
| Bike | 6 | 1/6 | 60° |
| Car | 3 | 1/12 | 30° |
| Total | 36 | 1 | 360° |
The total row closes the table. If the total angle is not 360°, a fraction was calculated wrongly or a category is missing.
How to draw the sectors with a protractor
Draw a circle with a compass and mark its center. Draw a vertical radius from the center straight up; this is the 0° line and the start of the first sector. Place the protractor with its center on the circle’s center and its 0° mark on that line, measure the first angle clockwise, and draw the closing radius.
That radius is the start of the next sector, so each measurement begins where the previous one ended, not at the 0° line. When you reach the last sector, its closing radius must land on the starting line. If it does not, add up the angles you measured to see where the difference built up.
Without a protractor, the folds of the paper will do: folding the circle in half marks 180°, and folding again marks 90°. The guide on fractions of a turn, degrees and radians collects the most common equivalences, such as 45° for an eighth. If you would rather shade a circle that is already divided into equal parts, the visual fractions tool draws it.
Worked example: how 36 students get to school
In a class of 36 students, 15 come by bus, 12 walk, 6 ride a bike and 3 come by car. Find the angle of each sector.
- Add the counts. 15 + 12 + 6 + 3 = 36, the total.
- Turn each count into a fraction and simplify. 15/36 = 5/12, 12/36 = 1/3, 6/36 = 1/6 and 3/36 = 1/12.
- Multiply each fraction by 360°. 5/12 × 360 = 150, 1/3 × 360 = 120, 1/6 × 360 = 60 and 1/12 × 360 = 30.
- Add the angles. 150 + 120 + 60 + 30 = 360.
The angles are 150°, 120°, 60° and 30°.
Check. The four fractions add up to 5/12 + 4/12 + 2/12 + 1/12 = 12/12 = 1, one full turn.
When the degrees are not whole numbers
If the total divides 360, each unit is worth a whole number of degrees and all the angles are whole. If it does not, some angles can still be whole. With 16 answers split 8, 4, 2 and 2, 360 ÷ 16 = 22.5° per answer, but the sectors measure 180°, 90°, 45° and 45°. The test is to look at each simplified fraction: the angle is whole when its denominator divides 360.
With 7 votes split 3, 2 and 2, none is whole, since 3/7 × 360 = 1080/7 and 2/7 × 360 = 720/7. In decimals, 1080/7 ≈ 154.29° and 720/7 ≈ 102.86°, rounded to two decimal places, with an error smaller than 0.005°. Because they are repeating decimals, they cannot be written out in full. The exact sum is 1080/7 + 720/7 + 720/7 = 2520/7 = 360.
Rounding to one decimal gives 154.3°, 102.9° and 102.9°, which add up to 360.1°, so the circle does not close. One possible fix is to subtract that tenth from one of the 102.9° sectors, so that 154.3° + 102.8° + 102.9° = 360.0°. You can also round to whole degrees, 154° + 103° + 103° = 360°, which is what a school protractor can mark. Keep the exact fraction until the last step; the round fractions tool gives the approximation with the error shown.
When to use bars instead of a pie chart
A pie chart works with a few categories that form a whole, especially if one of them takes up half a turn or a quarter, because those marks are recognized at a glance. With many categories or similar values, two sectors of 36° and 40° are almost impossible to tell apart. In a bar chart those differences show up as lengths on a common scale.
Wikipedia notes that comparing sizes is harder in a pie chart, and that comparison by angle has been found to be less accurate than comparison by length. The fraction of each category is calculated the same way in both charts, and the relative frequency of each one is the same division of the count by the total, as the guide on relative frequency explains. If the data are proportions of a split, the proportional sharing tool calculates each share.
Sources
Wikipedia, article Pie chart: a quantity that is a fraction Q of the total has a central angle of 360Q degrees, and comparing sizes is harder than in other charts, with comparison by angle less accurate than by length (accessed October 17, 2026).