Every Slice of a Pie Chart Is an Angle Out of 360
Learn to turn each share of data into a sector angle, read a quantity back off a given pie chart, test inverse proportion with a constant product, and solve worker-and-time or speed-and-time problems.
How do you turn a share of data into a slice of a circle?
By working out what fraction of 360 degrees that share deserves. If 15 students out of 40 come by bus, the bus sector is
A full circle is , so every category simply claims its fraction of it. This page covers everything in the CBSE Class 8 Mathematics chapter's second part: constructing a pie chart, reading one, testing inverse proportion, and inverse proportion problems.
A full circle is , so every category simply claims its fraction of it. This page covers everything in the CBSE Class 8 Mathematics chapter's second part: constructing a pie chart, reading one, testing inverse proportion, and inverse proportion problems.
Formula
How do you construct a pie chart from data?
Convert each category's share into a sector angle:
Worked example. Forty students report how they travel to school: bus , cycle , walk , car .
The check that must always be run:
Drawing it:
1. Draw a circle with compasses and mark its centre.
2. Draw one radius as a starting line.
3. Measure the first angle at the centre with a protractor and draw the next radius.
4. Measure each following angle from the previous radius, not from the starting line.
5. Label every sector with its category, and shade them differently or add a key.
A second example. A family's monthly spending of ₹24000: rent ₹8000, food ₹6000, school ₹4000, travel ₹3000, savings ₹3000. The angles are
adding to .
The total of 360 is the self-check, and it catches almost every arithmetic slip. If your angles add to 358 or 362, one fraction was computed wrongly — and drawing them without checking leaves a visible gap or overlap in the finished circle.
Worked example. Forty students report how they travel to school: bus , cycle , walk , car .
The check that must always be run:
Drawing it:
1. Draw a circle with compasses and mark its centre.
2. Draw one radius as a starting line.
3. Measure the first angle at the centre with a protractor and draw the next radius.
4. Measure each following angle from the previous radius, not from the starting line.
5. Label every sector with its category, and shade them differently or add a key.
A second example. A family's monthly spending of ₹24000: rent ₹8000, food ₹6000, school ₹4000, travel ₹3000, savings ₹3000. The angles are
adding to .
The total of 360 is the self-check, and it catches almost every arithmetic slip. If your angles add to 358 or 362, one fraction was computed wrongly — and drawing them without checking leaves a visible gap or overlap in the finished circle.
How do you read a quantity off a given pie chart?
Reverse the formula. A sector's share of the total is its angle out of 360:
Worked example. A pie chart shows the favourite sports of students, and the cricket sector measures :
Worked example. On the same chart a sector of represents
Worked example working backwards to the total. A sector of is known to represent people. Since is of the circle, the total is
As a percentage. A sector of is
of the whole, and one of is .
The angles worth recognising instantly are for a quarter, for a half, for a third and for a tenth.
The common error is treating the angle as the quantity itself. A sector does not mean 90 students — it means a quarter of however many there are, which was 180 here. So the total must always be brought into the calculation.
Worked example. A pie chart shows the favourite sports of students, and the cricket sector measures :
Worked example. On the same chart a sector of represents
Worked example working backwards to the total. A sector of is known to represent people. Since is of the circle, the total is
As a percentage. A sector of is
of the whole, and one of is .
The angles worth recognising instantly are for a quarter, for a half, for a third and for a tenth.
The common error is treating the angle as the quantity itself. A sector does not mean 90 students — it means a quarter of however many there are, which was 180 here. So the total must always be brought into the calculation.
How do you test whether two quantities are inversely proportional?
Multiply each pair of corresponding values and check that the product stays constant. In direct proportion the quotient is constant; in inverse proportion it is the product.
Worked example. Workers against days taken:
- workers, days
- workers, days
- workers, days
- workers, days
Multiplying each pair:
The product is constant at 180 worker-days, so the two are inversely proportional — more workers means fewer days.
Worked example. Speed against time for a fixed journey:
- km/h, hours
- km/h, hours
- km/h, hours
Constant at km, which is the distance — so inversely proportional.
In inverse proportion, doubling one halves the other. Going from 60 to 120 km/h halved the time from 4 hours to 2.
The pair often confused is speed and distance against speed and time. For a fixed journey, speed and time are inversely related, while at a fixed speed, distance and time are directly related — so the same three quantities behave both ways depending on which one is held constant.
Worked example. Workers against days taken:
- workers, days
- workers, days
- workers, days
- workers, days
Multiplying each pair:
The product is constant at 180 worker-days, so the two are inversely proportional — more workers means fewer days.
Worked example. Speed against time for a fixed journey:
- km/h, hours
- km/h, hours
- km/h, hours
Constant at km, which is the distance — so inversely proportional.
In inverse proportion, doubling one halves the other. Going from 60 to 120 km/h halved the time from 4 hours to 2.
The pair often confused is speed and distance against speed and time. For a fixed journey, speed and time are inversely related, while at a fixed speed, distance and time are directly related — so the same three quantities behave both ways depending on which one is held constant.
How do you solve an inverse proportion word problem?
Find the constant product first, then divide by the new value.
Workers and days. If workers build a wall in days, how long do workers take?
Provisions. A hostel has food for students for days. If more students join, how long will it last?
Speed and time. A bus covers a route in hours at km/h. How long at km/h?
Pipes filling a tank. If pipes fill a tank in hours, then pipes take
Finding the number needed. A job takes days with workers. To finish in days:
The direction of the change is the built-in check. More workers must mean fewer days, and more students must mean the food lasts a shorter time — so 18 workers taking 10 days rather than 22, and 50 students getting 24 days rather than 37, are both moving the right way. An answer that moves the wrong way means the multiplication and division were swapped.
Workers and days. If workers build a wall in days, how long do workers take?
Provisions. A hostel has food for students for days. If more students join, how long will it last?
Speed and time. A bus covers a route in hours at km/h. How long at km/h?
Pipes filling a tank. If pipes fill a tank in hours, then pipes take
Finding the number needed. A job takes days with workers. To finish in days:
The direction of the change is the built-in check. More workers must mean fewer days, and more students must mean the food lasts a shorter time — so 18 workers taking 10 days rather than 22, and 50 students getting 24 days rather than 37, are both moving the right way. An answer that moves the wrong way means the multiplication and division were swapped.
Exam tip
Exam tip: checking that your sector angles total 360
Pie chart and proportion questions each carry one decisive check.
After computing sector angles, add them and confirm exactly . Write that total in your working — examiners look for it, and it catches nearly every slip.
Measure each angle from the previous radius, and label every sector or supply a key.
When reading a chart, remember a sector angle is a fraction of 360, not the quantity itself — multiply by the total.
Before solving a proportion problem, say which type it is and why: inverse, since more workers means fewer days. Then find the constant product and divide.
And sense-check the direction of your answer: more workers, fewer days; higher speed, less time; more people, food lasts less long.
After computing sector angles, add them and confirm exactly . Write that total in your working — examiners look for it, and it catches nearly every slip.
Measure each angle from the previous radius, and label every sector or supply a key.
When reading a chart, remember a sector angle is a fraction of 360, not the quantity itself — multiply by the total.
Before solving a proportion problem, say which type it is and why: inverse, since more workers means fewer days. Then find the constant product and divide.
And sense-check the direction of your answer: more workers, fewer days; higher speed, less time; more people, food lasts less long.
Did you know
Why does scaling the time give the same answer as dividing the product?
Because the two routes are the same arithmetic written differently.
Going from 12 workers to 18 multiplies the workforce by . In inverse proportion the time is divided by that same factor, so it becomes
exactly matching the found from the constant product. Dividing the product by a larger number is scaling the old time down by the ratio of the workforces — which makes the scaling route a useful shortcut whenever the ratio divides cleanly.
Going from 12 workers to 18 multiplies the workforce by . In inverse proportion the time is divided by that same factor, so it becomes
exactly matching the found from the constant product. Dividing the product by a larger number is scaling the old time down by the ratio of the workforces — which makes the scaling route a useful shortcut whenever the ratio divides cleanly.
Key takeaways
Pie charts and inverse proportion: quick revision
- — so 15 out of 40 gives , and the angles must add to exactly 360.
- Measure each angle from the previous radius, and label every sector.
- To read a chart, — a sector of 720 students is 180, not 90.
- Recognise as a quarter, a third, a half and a tenth.
- Direct proportion has a constant quotient; inverse proportion has a constant product — worker-days.
- Solve inverse problems by finding the product and dividing: 18 workers take 10 days, 50 students get 24 days, and 80 km/h takes 3 hours — always checking the change runs the right way.
You will remember all of this far better after answering five questions on it than after reading it twice.
- Measure each angle from the previous radius, and label every sector.
- To read a chart, — a sector of 720 students is 180, not 90.
- Recognise as a quarter, a third, a half and a tenth.
- Direct proportion has a constant quotient; inverse proportion has a constant product — worker-days.
- Solve inverse problems by finding the product and dividing: 18 workers take 10 days, 50 students get 24 days, and 80 km/h takes 3 hours — always checking the change runs the right way.
You will remember all of this far better after answering five questions on it than after reading it twice.