Plane Figures
Pointwise AISSEE Class 6 Mathematics guide for Plane Figures (Topic 20). Master polygons, regular polygons, triangle angle sum property, triangle inequality, and quadrilateral classifications.
Overview & Fundamental Concept
Pointwise AISSEE Class 6 Mathematics guide for Plane Figures (Topic 20). Master polygons, regular polygons, triangle angle sum property, triangle inequality, and quadrilateral classifications.
Official Syllabus Coverage Breakdown
| Syllabus Item | Prescribed Scope | Exam Relevance & Pedagogical Objective |
|---|---|---|
| Item 20 | Classification of polygons, regular polygons, triangles (by sides and angles), quadrilaterals (parallelogram, trapezium, rhombus), angle sum property. | Master 2D polygon taxonomy, regular polygon angles, triangle angle sum / inequality laws, and quadrilateral hierarchy (parallelogram, rhombus, trapezium) for AISSEE Class VI. |
1. Polygon Classification & Regular Polygons
- Definition of Polygon: A simple closed 2D plane figure formed entirely by three or more straight line segments. The line segments are called sides, and their intersection points are called vertices.
- Polygon Names by Number of Sides: 3 sides = Triangle; 4 = Quadrilateral; 5 = Pentagon; 6 = Hexagon; 7 = Heptagon; 8 = Octagon; 9 = Nonagon; 10 = Decagon.
- Regular Polygon: A polygon in which all sides are equal in length and all interior angles are equal in measure (e.g., Equilateral triangle, Square, Regular Pentagon).
- Diagonal Formula for $n$-sided Polygon: $$\text{Number of Diagonals} = \frac{n(n - 3)}{2}$$ For triangle ($n = 3$): $3(0)/2 = 0$ diagonals. For quadrilateral ($n = 4$): $4(1)/2 = 2$ diagonals. For pentagon ($n = 5$): $5(2)/2 = 5$ diagonals.
| Polygon Name | Sides ($n$) | Interior Angle Sum: $(n - 2) \times 180^\circ$ | Number of Diagonals: $\frac{n(n-3)}{2}$ |
|---|---|---|---|
| Triangle | 3 | $(3 - 2) \times 180^\circ = 180^\circ$ | $\frac{3(0)}{2} = 0$ |
| Quadrilateral | 4 | $(4 - 2) \times 180^\circ = 360^\circ$ | $\frac{4(1)}{2} = 2$ |
| Pentagon | 5 | $(5 - 2) \times 180^\circ = 540^\circ$ | $\frac{5(2)}{2} = 5$ |
| Hexagon | 6 | $(6 - 2) \times 180^\circ = 720^\circ$ | $\frac{6(3)}{2} = 9$ |
| Octagon | 8 | $(8 - 2) \times 180^\circ = 1080^\circ$ | $\frac{8(5)}{2} = 20$ |
2. Triangles: Classifications, Angle Sum & Inequality Laws
- Classification by Sides:
• Equilateral: All 3 sides equal; each interior angle $= 60^\circ$.
• Isosceles: 2 sides equal; angles opposite to equal sides are equal.
• Scalene: All 3 sides of different lengths; all 3 angles different. - Classification by Angles:
• Acute-angled: All three angles are acute ($< 90^\circ$).
• Right-angled: Exactly one angle is $90^\circ$. By Pythagoras theorem: $h^2 = p^2 + b^2$.
• Obtuse-angled: Exactly one angle is obtuse ($> 90^\circ$). - Angle Sum Property of a Triangle: The sum of all three interior angles of any triangle is ALWAYS $180^\circ$: $$\angle A + \angle B + \angle C = 180^\circ$$
- Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be STRICTLY GREATER than the length of the third side: $$a + b > c, \quad b + c > a, \quad c + a > b$$
AISSEE Trap Alert • The Triangle Existence Test
Can a triangle have sides 4 cm, 5 cm, and 9 cm? Test: 4 + 5 = 9. Since 9 is NOT strictly greater than 9, NO triangle can be formed! Sum of two smaller sides MUST exceed the third side.
3. Quadrilaterals: Parallelogram, Rhombus, Trapezium & Square
- Angle Sum Property of Quadrilateral: The sum of all four interior angles of any quadrilateral is ALWAYS $360^\circ$.
- Parallelogram: Opposite sides are parallel and equal; opposite angles are equal; adjacent angles are supplementary ($180^\circ$); diagonals bisect each other.
- Rectangle: A parallelogram with all 4 interior angles equal to $90^\circ$. Diagonals are equal and bisect each other.
- Rhombus: A parallelogram with all 4 sides equal. Diagonals bisect each other perpendicularly at right angles ($90^\circ$).
- Square: A regular quadrilateral with all 4 sides equal and all 4 angles equal to $90^\circ$. Diagonals are equal and bisect each other at $90^\circ$.
- Trapezium: A quadrilateral with exactly one pair of parallel sides. (If non-parallel sides are equal, it is an Isosceles Trapezium).
| Quadrilateral | Opposite Sides Parallel? | All Sides Equal? | All Angles 90°? | Diagonals Bisect at 90°? |
|---|---|---|---|---|
| Parallelogram | Both pairs parallel | No | No | No |
| Rectangle | Both pairs parallel | No | Yes | No |
| Rhombus | Both pairs parallel | Yes | No | Yes (Perpendicular) |
| Square | Both pairs parallel | Yes | Yes | Yes (Perpendicular) |
| Trapezium | Only 1 pair parallel | No | No | No |
Cadet Mnemonic • Diagonals at 90° Anchor
Remember which shapes have diagonals bisecting at 90°:
- Rhombus • Square (Both have ALL 4 sides equal!)
- Equal sides = Perpendicular diagonals ($90^\circ$)
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: Exterior Angle Shortcut
An exterior angle of a triangle equals the sum of the two opposite interior angles: $\angle \text{Ext} = \angle 1 + \angle 2$.
Shortcut Technique 2: Regular Polygon Interior Angle
Each interior angle of a regular $n$-gon is $\frac{(n - 2) \times 180^\circ}{n}$. For regular hexagon: $\frac{4 \times 180}{6} = 120^\circ$.
Shortcut Technique 3: Adjacent Angles in Parallelogram
In any parallelogram, adjacent angles are supplementary: $\angle A + \angle B = 180^\circ$. If one angle is $70^\circ$, adjacent is $110^\circ$.
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 Plane Figures?
CADET MEMORY KEY
Interior Angle Sum: S_i = (n - 2) \times 180^\circ
REVISION FLASHCARD
Cadet Shortcut: How to solve Plane Figures 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
Triangle Angle Sum: Sum of 3 angles of any triangle is ALWAYS 180°.
REVISION FLASHCARD
Rapid Memory Anchor for AISSEE Hall Recall:
CADET MEMORY KEY
Quadrilateral Angle Sum: Sum of 4 angles of any quadrilateral is ALWAYS 360°.
REVISION FLASHCARD
How many diagonals does a regular hexagon have?
CADET MEMORY KEY
Formula: n(n − 3) / 2. For hexagon (n = 6): 6 × (6 − 3) / 2 = (6 × 3) / 2 = 18 / 2 = 9 diagonals.
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
Triangle Angle Sum
$$\angle A + \angle B + \angle C = 180^\circ$$
Interior angle sum of any triangle.
Polygon Interior Angle Sum
$$\sum = (n - 2) \times 180^\circ$$
Total interior sum for n-sided polygon.
Number of Diagonals
$$D = \frac{n(n - 3)}{2}$$
Total diagonal count for n-sided polygon.
Step-by-Step Solved Examination Models
Solved Example 1
Model Problem
Two angles of a triangle are $55^\circ$ and $65^\circ$. Find the third angle and classify the triangle by angles.
Detailed Solution:
Angle sum property of triangle: Sum = $180^\circ$.
Third angle = $180^\circ - (55^\circ + 65^\circ) = 180^\circ - 120^\circ = 60^\circ$.
Since all three angles ($55^\circ, 65^\circ, 60^\circ$) are $< 90^\circ$, it is an acute-angled triangle.
Solved Example 2
Model Problem
Three angles of a quadrilateral are $85^\circ, 95^\circ,$ and $110^\circ$. Find the fourth angle.
Detailed Solution:
Sum of interior angles of a quadrilateral = $360^\circ$.
Fourth angle = $360^\circ - (85^\circ + 95^\circ + 110^\circ) = 360^\circ - 290^\circ = 70^\circ$.
Solved Example 3
Model Problem
In a triangle, the angles are in the ratio 2 : 3 : 4. Find the measure of the smallest angle and the largest angle.
Detailed Solution:
1. Let the angles of the triangle be $2x, 3x,$ and $4x$.
2. By Angle Sum Property of a triangle: $2x + 3x + 4x = 180^\circ$.
3. $9x = 180^\circ \implies x = 20^\circ$.
4. Smallest angle $= 2x = 2 \times 20^\circ = 40^\circ$.
5. Largest angle $= 4x = 4 \times 20^\circ = 80^\circ$.
Middle angle $= 3x = 60^\circ$. Check: $40 + 60 + 80 = 180^\circ$. Correct Option is (A).
2. By Angle Sum Property of a triangle: $2x + 3x + 4x = 180^\circ$.
3. $9x = 180^\circ \implies x = 20^\circ$.
4. Smallest angle $= 2x = 2 \times 20^\circ = 40^\circ$.
5. Largest angle $= 4x = 4 \times 20^\circ = 80^\circ$.
Middle angle $= 3x = 60^\circ$. Check: $40 + 60 + 80 = 180^\circ$. Correct Option is (A).
AISSEE High-Yield Practice Drills (10 Questions)
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Question 1 of 10
Exam Standard
What is the sum of the interior angles of any triangle?
Correct Answer: Option (A)
The angle sum property of Euclidean triangles states that the sum of the interior angles is always $180^\circ$.
Question 2 of 10
Exam Standard
A triangle having all three sides of equal length is called:
Correct Answer: Option (A)
An equilateral triangle has 3 equal sides and each interior angle measures $60^\circ$.
Question 3 of 10
Exam Standard
A polygon with 5 sides is called a:
Correct Answer: Option (A)
A 5-sided polygon is a pentagon.
Question 4 of 10
Exam Standard
In an isosceles triangle, if the vertical angle is $80^\circ$, find each base angle.
Correct Answer: Option (A)
Base angles of an isosceles triangle are equal: $(180^\circ - 80^\circ) / 2 = 100^\circ / 2 = 50^\circ$.
Question 5 of 10
Exam Standard
What is the measure of each interior angle of an equilateral triangle?
Correct Answer: Option (A)
$180^\circ / 3 = 60^\circ$.
Question 6 of 10
Exam Standard
Can a triangle have sides measuring 3 cm, 4 cm, and 8 cm?
Correct Answer: Option (A)
Triangle Inequality Theorem: The sum of the lengths of any two sides must be strictly greater than the third side. Here $3 + 4 = 7 < 8$, which violates the rule.
Question 7 of 10
Exam Standard
A quadrilateral whose opposite sides are parallel and equal, but diagonals are NOT equal, is called a:
Correct Answer: Option (A)
In a general parallelogram, opposite sides are parallel and equal, but diagonals are unequal (unless it is a rectangle).
Question 8 of 10
Exam Standard
What is the sum of the interior angles of a quadrilateral?
Correct Answer: Option (A)
A quadrilateral can be split into 2 triangles: $2 \times 180^\circ = 360^\circ$.
Question 9 of 10
Exam Standard
In a right-angled triangle, if one acute angle is $35^\circ$, find the other acute angle.
Correct Answer: Option (A)
The two acute angles in a right-angled triangle are complementary: $90^\circ - 35^\circ = 55^\circ$.
Question 10 of 10
Exam Standard
A quadrilateral having only ONE pair of opposite sides parallel is a:
Correct Answer: Option (A)
By definition, a quadrilateral with exactly one pair of parallel sides is a trapezium.
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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 |
|---|---|
| Triangle Angle Sum | Sum of angles in any triangle is strictly 180°. |
| Quad Angle Sum | Sum of angles in any quadrilateral is strictly 360°. |
| Triangle Inequality | Sum of any two sides must be strictly greater than the third side. |
| Rhombus Diagonals | Diagonals of a rhombus bisect each other at right angles (90°). |
Frequently Asked Questions
How many questions from Plane Figures appear in the AISSEE Class VI exam?
Typically, 2 to 3 questions directly test Plane Figures 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 Plane Figures before the exam?
Review the high-yield formulas, practice the 10 interactive drill MCQs on this page, and memorize the revision checkpoints matrix.