Circle
Pointwise AISSEE Class 6 Mathematics guide for Circle (Topic 17). Master center, radius, diameter, longest chord, tangent, secant, arc, sector, segment, circumference (2πr), and area (πr²).
Overview & Fundamental Concept
Pointwise AISSEE Class 6 Mathematics guide for Circle (Topic 17). Master center, radius, diameter, longest chord, tangent, secant, arc, sector, segment, circumference (2πr), and area (πr²).
Official Syllabus Coverage Breakdown
| Syllabus Item | Prescribed Scope | Exam Relevance & Pedagogical Objective |
|---|---|---|
| Item 17 | Center, radius, diameter, chord, secant, tangent, arc, sector, segment, circumference, and relation between radius and diameter. | Master circle geometry definitions, center-radius-diameter relations, chord vs secant vs tangent differences, sector vs segment, and circumference/area calculations for AISSEE Class VI. |
1. Primary Anatomy: Center, Radius, Diameter & Chords
- Definition of Circle: A circle is a closed two-dimensional plane figure consisting of all points that are equidistant from a fixed interior point called the Center ($O$).
- Radius ($r$): A straight line segment connecting the center of the circle to any point on its circumference. All radii of a given circle are strictly equal in length.
- Diameter ($d$): A straight line segment passing through the center of the circle and joining two points on the circumference: $$d = 2r \iff r = \frac{d}{2}$$
- Longest Chord Property: A chord is a line segment connecting any two points on a circle. The diameter is the LONGEST chord of a circle. Every diameter is a chord, but not every chord is a diameter.
- Concentric Circles: Circles having the same center but different radii.
| Geometric Element | Definition | Key Mathematical Relation |
|---|---|---|
| Radius ($r$) | Line segment from Center to Circumference | $r = d / 2$ |
| Diameter ($d$) | Line segment through Center joining two boundary points | $d = 2r$ (Longest chord) |
| Chord | Line segment joining any two points on circle | Length $\le 2r$ |
| Circumference ($C$) | Total perimeter length of boundary | $C = 2\pi r = \pi d$ |
| Area ($A$) | Enclosed surface space | $A = \pi r^2$ |
Cadet Mnemonic • D-R Relationship Anchor
Remember diameter is double radius:
- Diameter = Double the radius ($d = 2r$)
- Radius = Reduced by half ($r = d / 2$)
2. Boundary Lines: Chord, Secant & Tangent
- Chord: A line segment having both its endpoints lying on the circle (finite segment).
- Secant: A straight line that passes through the circle, cutting it at two distinct points (infinite line extending outside).
- Tangent: A straight line in the plane that touches the circle at exactly one single point (called the point of contact).
- Tangent-Radius Perpendicularity: The radius drawn to the point of contact of a tangent is always perpendicular ($90^\circ$) to the tangent line.
| Element | Nature of Figure | Points of Contact / Intersection with Circle |
|---|---|---|
| Chord | Line Segment | 2 endpoints on the boundary |
| Secant | Line (infinite) | Intersects at 2 distinct points |
| Tangent | Line (infinite) | Touches at exactly 1 point |
AISSEE Trap Alert • Chord vs Secant vs Tangent
Remember: A chord is a SEGMENT (ends on circle). A secant is a LINE (crosses through). A tangent is a LINE that TOUCHES at only ONE point!
3. Circular Regions: Arc, Sector & Segment
- Arc: Any continuous curved portion of the circumference. Minor arc ($< \text{semicircle}$) vs Major arc ($> \text{semicircle}$).
- Semicircle: Half of a circle formed by a diameter. An angle inscribed in a semicircle is ALWAYS a right angle ($90^\circ$).
- Sector: The interior region enclosed by two radii and an arc (resembles a slice of pizza).
- Segment: The interior region enclosed by a chord and an arc.
- Circumference & Area Formulas:
• $\text{Circumference } C = 2\pi r = \pi d$
• $\text{Area } A = \pi r^2$
(Use $\pi \approx \frac{22}{7}$ unless specified otherwise).
Exam Trick • Sector vs Segment Distinction
Sector starts with Slices of pizza (bounded by 2 Radii). Segment has a Chord (looks like a bow and string)!
Visual Concept Blueprint • Vector Architecture
Speed Shortcuts & Mental Calculation Strategies
Master these high-yield time-saving heuristics curated by top AISSEE scorers to eliminate calculation bottlenecks and answer questions in under 45 seconds.
Shortcut Technique 1: Radius Halving First Step
If diameter is given in an area problem (e.g., d = 14 cm), always divide by 2 first (r = 7 cm) before squaring: $A = \frac{22}{7} \times 7 \times 7 = 154\text{ cm}^2$.
Shortcut Technique 2: Circumference to Radius Shortcut
For $\pi = 22/7$, if Circumference is 44 cm, Radius is 7 cm. If Circumference is 88 cm, Radius is 14 cm (Multiples of 44 map to multiples of 7).
Shortcut Technique 3: Semicircle Perimeter Trap
The perimeter of a semicircle is NOT half the circumference! It is half circumference PLUS the diameter: $P = \pi r + 2r$.
Interactive 3D Revision Flashcards Deck
Click any card to flip and reveal the key exam rule or topper secret.
REVISION FLASHCARD
What is the core rule & definition for Circle?
CADET MEMORY KEY
Diameter Relation: d = 2r \implies r = \frac{d}{2}
REVISION FLASHCARD
Cadet Shortcut: How to solve Circle in under 30 seconds?
CADET MEMORY KEY
Fast MCQ Elimination: Check the last digit (unit digit) and estimate the order of magnitude to eliminate 2 distractors in under 10 seconds.
REVISION FLASHCARD
Exam Trap Alert: What pitfall do most candidates fall into?
CADET MEMORY KEY
Longest Chord: The diameter is the longest chord that can be drawn in any circle.
REVISION FLASHCARD
Rapid Memory Anchor for AISSEE Hall Recall:
CADET MEMORY KEY
Sector vs Segment: Sector is bounded by two radii and an arc (pizza slice). Segment is bounded by a chord and an arc.
REVISION FLASHCARD
What is the perimeter of a semicircle of radius 7 cm?
CADET MEMORY KEY
Perimeter = πr + 2r = (22/7 × 7) + (2 × 7) = 22 + 14 = 36 cm.
Common Traps & Examination Pitfalls
Avoid the Speed-Guess Trap: Under AISSEE pressure, candidates frequently misread units (e.g., mixing metres and centimetres, or per hour and per second) and confuse complementary ($90^\circ$) with supplementary ($180^\circ$) angles. Read every stem carefully and check units before selecting your OMR option.
Key Identities & Formulas
Diameter-Radius Relation
$$d = 2r \iff r = \frac{d}{2}$$
Fundamental proportionality.
Circumference
$$C = 2\pi r = \pi d$$
Total boundary length of circle.
Area of Circle
$$A = \pi r^2 = \frac{\pi d^2}{4}$$
Total enclosed planar area.
Step-by-Step Solved Examination Models
Solved Example 1
Model Problem
The radius of a circular playground is 14 m. Find its circumference and area (Take $\pi = 22/7$).
Detailed Solution:
Circumference = $2 \pi r = 2 \times \frac{22}{7} \times 14 = 88$ m.
Area = $\pi r^2 = \frac{22}{7} \times 14 \times 14 = 22 \times 28 = 616\text{ sq m}$.
Solved Example 2
Model Problem
What is the longest chord of a circle of radius 8.5 cm?
Detailed Solution:
The longest chord of any circle is its diameter: $\text{Diameter} = 2 \times \text{radius} = 2 \times 8.5 = 17$ cm.
Solved Example 3
Model Problem
The circumference of a circular cricket ground is 176 meters. Find the area of the ground. (Take $\pi = 22/7$)
Detailed Solution:
1. Circumference $C = 2\pi r = 176\text{ m}$.
2. $2 \times \frac{22}{7} \times r = 176 \implies \frac{44}{7} \times r = 176$.
3. $r = \frac{176 \times 7}{44} = 4 \times 7 = 28\text{ meters}$.
4. $\text{Area } A = \pi r^2 = \frac{22}{7} \times 28 \times 28$.
5. $A = 22 \times 4 \times 28 = 88 \times 28 = 2,464\text{ m}^2$. Correct Option is (A).
2. $2 \times \frac{22}{7} \times r = 176 \implies \frac{44}{7} \times r = 176$.
3. $r = \frac{176 \times 7}{44} = 4 \times 7 = 28\text{ meters}$.
4. $\text{Area } A = \pi r^2 = \frac{22}{7} \times 28 \times 28$.
5. $A = 22 \times 4 \times 28 = 88 \times 28 = 2,464\text{ m}^2$. Correct Option is (A).
AISSEE High-Yield Practice Drills (10 Questions)
Simulated exam drill matching NTA question style, difficulty, and phrasing. Click an option for instant feedback and pedagogical explanation.
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Cadet Streak 🔥
Question 1 of 10
Exam Standard
If the diameter of a circle is 28 cm, find its radius.
Correct Answer: Option (A)
Radius = $\text{Diameter} / 2 = 28 / 2 = 14$ cm.
Question 2 of 10
Exam Standard
Find the circumference of a circle whose radius is 21 cm (Use $\pi = 22/7$).
Correct Answer: Option (A)
$C = 2 \times \frac{22}{7} \times 21 = 2 \times 22 \times 3 = 132$ cm.
Question 3 of 10
Exam Standard
A line segment joining the center of a circle to any point on the boundary is called:
Correct Answer: Option (A)
By geometric definition, the line segment connecting the center to any point on the circle is the radius.
Question 4 of 10
Exam Standard
A line segment connecting any two points on a circle is called:
Correct Answer: Option (A)
Any straight line segment whose endpoints both lie on the circular boundary is a chord.
Question 5 of 10
Exam Standard
How many diameters can be drawn in a circle?
Correct Answer: Option (A)
Through the center of a circle, infinitely many straight lines can pass, creating infinite diameters.
Question 6 of 10
Exam Standard
The distance around the boundary of a circular running track is called its:
Correct Answer: Option (A)
The total linear perimeter of a circle is termed its circumference.
Question 7 of 10
Exam Standard
If the circumference of a circular plate is 44 cm, find its radius (Take $\pi = 22/7$).
Correct Answer: Option (A)
$2 \times \frac{22}{7} \times r = 44 \implies \frac{44}{7} r = 44 \implies r = 7$ cm.
Question 8 of 10
Exam Standard
What is half of a full circle called?
Correct Answer: Option (A)
A diameter divides the interior and boundary of a circle into two equal halves called semicircles.
Question 9 of 10
Exam Standard
The region bounded by two radii and an arc of a circle is called a:
Correct Answer: Option (A)
A sector is a portion of a circle enclosed by two radii and the connecting arc (like a pizza slice).
Question 10 of 10
Exam Standard
A wheel of radius 35 cm makes 100 complete revolutions. What is the total distance covered?
Correct Answer: Option (A)
Distance in 1 revolution = $2 \times \frac{22}{7} \times 35 = 220$ cm. Total distance for 100 turns = $220 \times 100 = 22,000\text{ cm} = 220$ m.
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Last-Minute Cadet Revision Matrix
Rapid recall checkpoints to review 24 hours before stepping into the examination hall.
| Concept Checkpoint | High-Yield Exam Rule & Memory Anchor |
|---|---|
| Longest Chord | The diameter is the longest chord of a circle. |
| Diameter Formula | Diameter = 2 × Radius; Radius = Diameter ÷ 2. |
| Sector vs Segment | Sector is bounded by two radii and an arc; Segment is bounded by a chord and an arc. |
| Tangent Property | A tangent touches a circle at exactly one point and is perpendicular to the radius. |
Frequently Asked Questions
How many questions from Circle appear in the AISSEE Class VI exam?
Typically, 2 to 3 questions directly test Circle in the 50-question Mathematics paper, contributing 6 to 9 marks towards the 150-mark section total.
Is there any negative marking for incorrect answers in AISSEE Class VI?
No. The National Testing Agency (NTA) does not apply negative marking in the AISSEE Class VI entrance examination. Candidates should attempt every single question.
What is the best way to revise Circle before the exam?
Review the high-yield formulas, practice the 10 interactive drill MCQs on this page, and memorize the revision checkpoints matrix.