Circles: Formulas, Theorems and Solved Questions
Circle questions are common in geometry and mensuration. Students must know radius, diameter, circumference, area, chord, tangent and sector basics. This topic is useful for SSC, railway, banking, police, state exams, defence exams, teaching exams and placement aptitude tests. In recent competitive exam preparation, students need both concept and practice. So this tutorial gives definitions, formulas, shortcuts, solved examples and many practice questions.
The examples are written in competitive exam style. They are based on common patterns seen in current aptitude resources, mock tests and written exam preparation pages. They are not claimed as exact official previous-year questions. The aim is to help you understand the topic and solve similar questions faster.
Use this page as a practice sheet. First read the rule, then solve the example by yourself, then check the solution. After that solve the practice set with a timer. This method improves speed and accuracy together.
What is Circles?
Circles is an important maths topic because it appears directly in school exams and indirectly in competitive exam questions. Sometimes the question is direct, and sometimes it is mixed with percentage, ratio, simplification, algebra or geometry. If the basic idea is clear, the mixed questions also become easier.
Important subtopics
- Radius
- Diameter
- Circumference
- Area
- Chord
- Tangent
- Sector
- Arc
- Angle in circle
- Exam shortcuts
Rules and formulas
- Read the last line of the question first to know what is required.
- Write the given values in short form before calculation.
- Keep units same, especially in length, area, volume, time and speed questions.
- Use direct formula for basic questions and shortcut method for repeated patterns.
- For objective exams, estimate the answer once before detailed calculation.
- Check options to avoid unnecessary long calculation.
- After solving, verify whether the answer is reasonable.
Solved examples for Circles
Example 1
Question: Find circumference of circle with radius 7 cm.
Solution: Circumference = 2πr = 2×22/7×7 = 44 cm.
Example 2
Question: Find area of circle with radius 14 cm.
Solution: Area = πr² = 22/7×14×14 = 616 square cm.
Example 3
Question: Diameter is 20 cm. Find radius.
Solution: Radius = diameter/2 = 10 cm.
Example 4
Question: Find diameter if radius is 13 cm.
Solution: Diameter = 2×13 = 26 cm.
Competitive exam style solved examples
Example 5
Question: A candidate gets 156 out of 200 marks. Find percentage.
Answer: 78%.
Method: Identify the topic first, apply the direct formula, and check the unit or option. This is the safest approach in written exams because it reduces calculation mistakes.
Example 6
Question: Find 35% of 640.
Answer: 224.
Method: Identify the topic first, apply the direct formula, and check the unit or option. This is the safest approach in written exams because it reduces calculation mistakes.
Example 7
Question: Average of 14, 18, 22, 26 and 30 is what?
Answer: 22.
Method: Identify the topic first, apply the direct formula, and check the unit or option. This is the safest approach in written exams because it reduces calculation mistakes.
Example 8
Question: A sum of Rs 1440 is divided in 5:7. Find shares.
Answer: Rs 600 and Rs 840.
Method: Identify the topic first, apply the direct formula, and check the unit or option. This is the safest approach in written exams because it reduces calculation mistakes.
Example 9
Question: CP is Rs 900 and SP is Rs 1080. Find profit percent.
Answer: 20%.
Method: Identify the topic first, apply the direct formula, and check the unit or option. This is the safest approach in written exams because it reduces calculation mistakes.
Example 10
Question: Find SI on Rs 5000 at 9% for 2 years.
Answer: Rs 900.
Method: Identify the topic first, apply the direct formula, and check the unit or option. This is the safest approach in written exams because it reduces calculation mistakes.
Example 11
Question: A bus travels 240 km in 4 hours. Find speed.
Answer: 60 km/h.
Method: Identify the topic first, apply the direct formula, and check the unit or option. This is the safest approach in written exams because it reduces calculation mistakes.
Example 12
Question: A can complete a work in 10 days and B in 15 days. Find time together.
Answer: 6 days.
Method: Identify the topic first, apply the direct formula, and check the unit or option. This is the safest approach in written exams because it reduces calculation mistakes.
Example 13
Question: Find HCF of 54 and 72.
Answer: 18.
Method: Identify the topic first, apply the direct formula, and check the unit or option. This is the safest approach in written exams because it reduces calculation mistakes.
Example 14
Question: Find LCM of 12 and 30.
Answer: 60.
Method: Identify the topic first, apply the direct formula, and check the unit or option. This is the safest approach in written exams because it reduces calculation mistakes.
Practice questions with answers
- A candidate gets 156 out of 200 marks. Find percentage. Answer: 78%.
- Find 35% of 640. Answer: 224.
- Average of 14, 18, 22, 26 and 30 is what? Answer: 22.
- A sum of Rs 1440 is divided in 5:7. Find shares. Answer: Rs 600 and Rs 840.
- CP is Rs 900 and SP is Rs 1080. Find profit percent. Answer: 20%.
- Find SI on Rs 5000 at 9% for 2 years. Answer: Rs 900.
- A bus travels 240 km in 4 hours. Find speed. Answer: 60 km/h.
- A can complete a work in 10 days and B in 15 days. Find time together. Answer: 6 days.
- Find HCF of 54 and 72. Answer: 18.
- Find LCM of 12 and 30. Answer: 60.
- Solve 3x + 8 = 29. Answer: 7.
- Find next term: 5, 11, 23, 47, ? Answer: 95.
- Find median of 3, 9, 11, 15, 20. Answer: 11.
- Find range of 12, 18, 25, 31. Answer: 19.
- Find probability of getting an even number on a dice. Answer: 1/2.
More solved practice questions
Question 1
A number is increased by 20% and becomes 960.
Solution: Original number = 960 × 100 / 120 = 800.
Question 2
A value decreases from 1200 to 1020.
Solution: Decrease = 180. Decrease percent = 180/1200 × 100 = 15%.
Question 3
A shopkeeper gives 10% discount and sells an item for Rs 1350.
Solution: Marked price = 1350 × 100 / 90 = Rs 1500.
Question 4
A car covers 315 km in 5 hours 15 minutes.
Solution: 5 hours 15 minutes = 5.25 hours. Speed = 315/5.25 = 60 km/h.
Question 5
A rectangle has perimeter 86 cm and length 25 cm.
Solution: 2(l+b)=86, so l+b=43. Breadth = 18 cm.
Question 6
The average of six numbers is 19. One more number 31 is added.
Solution: Old total = 114. New total = 145. New average = 145/7 = 20.71.
Question 7
A worker completes 1/5 of work in one day.
Solution: Total time = 5 days.
Question 8
Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
Question 9
If x:y = 3:4 and y:z = 8:9, find x:z.
Solution: Make y common. x:y = 6:8 and y:z = 8:9. So x:z = 6:9 = 2:3.
Question 10
Find next term: 2, 5, 10, 17, 26, ?
Solution: Differences are 3,5,7,9. Next difference is 11. Answer = 37.
Shortcut tricks
For percentage, remember fraction conversions. 50% is 1/2, 25% is 1/4, 20% is 1/5, 12.5% is 1/8 and 10% is 1/10. This saves time in SSC, railway, banking and police exam calculations.
For formula questions, avoid memorising without examples. Write one solved example below each formula in your notebook. This helps you remember where the formula is used.
For geometry questions, draw a small rough figure. For number questions, check divisibility, place value and unit digit. For algebra questions, substitute simple values when checking an identity or comparison.
Common mistakes
- Using the wrong base in percentage questions.
- Forgetting to convert units before applying a formula.
- Using area formula when perimeter is asked.
- Not arranging values before finding median.
- Ignoring words like only, both, at least, total, difference and remaining.
- Doing long calculation when approximation or option elimination is enough.
- Not revising wrong questions after practice.
How to revise Circles for exams
Start with easy examples and solve them without a timer. After the basic method is clear, solve mixed questions. In the final stage, use a timer and try to solve questions in exam speed. Keep a small list of mistakes and revise it every two or three days.
Recent competitive exam questions often combine two ideas. For example, a decimals question may include percentage, a divisibility question may include remainder, and a geometry question may include ratio. So after learning this topic, solve mixed practice questions too.
FAQ on Circles
Is Circles important for competitive exams?
Yes. It is useful for maths, quantitative aptitude, numerical ability and reasoning-related questions.
How many examples should I solve?
Solve at least 30 to 50 examples from the topic. Then solve mixed questions with a timer.
Should I learn shortcuts first?
No. First learn the normal method. After that, shortcuts become easy and safe to use.
Are these exact previous-year questions?
No. These are competitive exam style questions based on common recent patterns. They are written for practice and understanding.
Conclusion
Circles becomes easy when formulas are learned with examples. Practice regularly, check mistakes and revise short methods before the exam.
Extra solved example
Question: If x:y = 3:4 and y:z = 8:9, find x:z.
Solution: Make y common. x:y = 6:8 and y:z = 8:9. So x:z = 6:9 = 2:3.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: The average of six numbers is 19. One more number 31 is added.
Solution: Old total = 114. New total = 145. New average = 145/7 = 20.71.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: A value decreases from 1200 to 1020.
Solution: Decrease = 180. Decrease percent = 180/1200 × 100 = 15%.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: A worker completes 1/5 of work in one day.
Solution: Total time = 5 days.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.
Extra solved example
Question: Find 12.5% of 1440.
Solution: 12.5% = 1/8. 1440/8 = 180.
This type of question is useful for competitive exams because it checks both concept and calculation speed. Try changing the numbers and solving the same pattern again.