19. Circles
Free study material · concepts, shortcuts & solved questions
Radius r is half the diameter. Circumference=2πr=πd and area=πr². The angle in a semicircle is 90°. Tangents are perpendicular to the radius at the point of contact; tangents from an external point to a circle are equal.
Method before formula Use π=22/7 when the numbers are multiples of 7 and the question permits it; otherwise use the stated value. Distinguish chord, diameter, secant and tangent. |
Formula and decision table
Term | Meaning |
Chord | Line segment joining two points on circle |
Diameter | Longest chord through centre |
Tangent | Touches circle at one point |
Secant | Cuts circle at two points |
Worked understanding
A reliable solution has five visible moves: identify the data, choose the base or unit, write the rule, substitute carefully, and check the size of the answer. The examples in this chapter are deliberately small so that the method remains visible.
Calculation discipline Before pressing ahead, state the denominator or unit in words. For example: profit on CP, discount on MP, average over number of items, speed in metres per second, and probability over the fixed sample space. |
Solved examples from this chapter
Question focus | Correct move | Answer |
For r=7, circumference using 22/7: | 2×22/7×7=44. | B. 44 |
For r=7, area: | 22/7×49=154. | C. 154 |
Angle in a semicircle is: | Thales’ theorem. | C. 90° |
What an examiner is testing
The options around Circles usually represent a wrong base, a missed conversion, a sign error or a shortcut used outside its condition. Say the base and unit before calculating, estimate the result, and use substitution or a reverse operation as the final check.
Step-by-step answer construction
Example 1: For r=7, circumference using 22/7: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is 44, because 2×22/7×7=44.
Example 2: For r=7, area: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is 154, because 22/7×49=154.
Example 3: Angle in a semicircle is: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is 90°, because Thales’ theorem.
Chapter practice
1. For r=7, circumference using 22/7:
(A) 22 (B) 44 (C) 88 (D) 154
Answer: B. 44 | Explanation: 2×22/7×7=44.
2. For r=7, area:
(A) 44 (B) 77 (C) 154 (D) 308
Answer: C. 154 | Explanation: 22/7×49=154.
3. Angle in a semicircle is:
(A) 45° (B) 60° (C) 90° (D) 180°
Answer: C. 90° | Explanation: Thales’ theorem.
SSC CGL speed and trap clinic
For Circles, speed comes after classification. Before calculating, say aloud what the number means, what the denominator is, and what unit the answer must carry. Then use the nearest-option check to catch a sign, base or conversion error.
Checkpoint | What to verify | Typical SSC mistake |
Base | Which quantity is the base in Circles? | Using selling price instead of cost price, or part instead of total |
Unit | Are time, distance, area and rate in compatible units? | Mixing hours with minutes or km/h with m/s |
Direction | Should the answer increase, decrease or stay bounded? | Accepting an impossible negative length or probability above 1 |
Estimate | What range should the answer lie in before exact work? | Trusting a long calculation that is far from the options |
Reverse check | Can the answer be substituted back into the condition? | Stopping at an algebraic value without verification |
Mini decision drill
1. Write the first operation you would perform in a Circles question and why.
2. State the most dangerous denominator or unit in this chapter.
3. Give one condition under which a shortcut would be invalid.
4. Create a small numerical example and verify it by a second method.
Revision evidence Do not tick this chapter because you recognised the formula. Tick it only after you solve one direct question, one altered-condition question and one mixed-paper question correctly. |