Notes For All Chapters – Maths Class 6 Ganita Prakash
📐 Perimeter and Area
Chapter 6 — Complete Study Notes with Formulas, Examples, and Practice Questions
🟩 Area
📏 Rectangle
⬛ Square
🔺 Triangle
🔢 Exam Ready
6.1 What is Perimeter?
The perimeter of a closed plane figure is the total distance covered along its boundary when you go around it once.
For a polygon (a closed figure made of straight line segments), the perimeter is simply the sum of all its sides.
Think of perimeter as putting a fence around a field, or putting lace around a tablecloth. The total length of the fence or lace = Perimeter!
A polygon is any closed figure made up of straight line segments — like triangle, square, rectangle, pentagon, hexagon, etc.
Perimeter of a Rectangle
📖 Understanding the Formula
A rectangle has 4 sides. The two longer sides are called the length (l) and the two shorter sides are called the breadth (b) or width.
Since opposite sides of a rectangle are equal: AB = CD (length) and BC = DA (breadth).
Opposite sides of a rectangle are always equal. So AB = CD and AD = BC.
🔢 Derivation
= AB + BC + AB + BC (opposite sides are equal)
= 2 × AB + 2 × BC
= 2 × (AB + BC)
= 2 × (length + breadth)
= 2(l + b)
📘 Worked Examples
Perimeter = 2 × (12 + 8) = 2 × 20 = 40 cm
Perimeter = 2 × (3 + 2) = 2 × 5 = 10 m
Perimeter of square park = 4 × 75 = 300 m
Distance in 3 rounds = 3 × 300 = 900 m
Perimeter of a Square
A square has all 4 sides equal. So instead of adding each side, we simply multiply the length of one side by 4.
= 4 × side
= 4s
Debojeet wants to put coloured tape all around a square photo frame of side 1 m.
Length of tape = Perimeter = 4 × 1 m = 4 m
The perimeter of a square is quadruple (four times) the length of its side.
🔄 Wire Bending Concept
If a rectangle is bent into a square (same wire = same perimeter), the side of the square can be found by dividing the perimeter by 4.
Perimeter of rectangle = 2 × (5 + 3) = 16 cm
Side of square = 16 ÷ 4 = 4 cm
Perimeter of a Triangle
A triangle has 3 sides. The perimeter is the sum of all three sides.
📐 Special Triangles
All 3 sides are equal.
Perimeter = 3 × side
Two sides are equal.
Perimeter = 2a + b
Perimeter = 4 + 5 + 7 = 16 cm
Third side = 55 − 20 − 14 = 21 cm
Both are regular polygons — all sides and all angles are equal. Perimeter = number of sides × length of one side.
Perimeter of a Regular Polygon
A regular polygon is a closed figure where all sides are equal AND all angles are equal. Examples: equilateral triangle, square, regular pentagon, regular hexagon.
📊 Formula Table
| Shape | Sides | Perimeter Formula |
|---|---|---|
| Equilateral Triangle | 3 | 3 × s |
| Square | 4 | 4 × s |
| Regular Pentagon | 5 | 5 × s |
| Regular Hexagon | 6 | 6 × s |
| Regular Octagon | 8 | 8 × s |
For any regular polygon: Perimeter = n × s, where n = number of sides and s = side length. As n increases, the shape gets closer and closer to a circle!
🔤 Straight and Diagonal Units
On dotted paper, lines can be of two types: straight lines (horizontal/vertical) and diagonal lines (slanted). These have different lengths!
Perimeter in mixed units = 6s + 3d means 6 straight units + 3 diagonal units. Straight and diagonal units are NOT the same length.
6.2 What is Area?
The area of a closed figure is the amount of region enclosed (surrounded) by that figure.
Area measures how much surface a figure covers.
📏 Unit of Area
Area is measured in square units. The standard units are:
Area of a square with side 1 cm
Area of a square with side 1 m
Squares can tile a surface completely without any gaps or overlaps. Circles cannot do this — they leave gaps. That’s why we use square units to measure area!
📐 Estimating Area on Graph Paper
To estimate the area of any shape using graph paper (1 square = 1 sq unit), follow these 4 rules:
- Count each complete square inside the shape as 1 sq unit.
- Ignore portions that are less than half a square.
- Count portions that are more than half a square as 1 sq unit.
- Count portions that are exactly half a square as ½ sq unit.
Area of Rectangle and Square
▭ Area of a Rectangle
If you place unit squares inside a rectangle of length 5 and breadth 3, you get 5 × 3 = 15 squares — that’s the area!
⬛ Area of a Square
📘 Worked Examples
Area of floor = 5 × 4 = 20 sq m
Area of carpet = 3 × 3 = 9 sq m
Uncarpeted area = 20 − 9 = 11 sq m
Area of whole land = 12 × 10 = 120 sq m
Area of one flower bed = 4 × 4 = 16 sq m
Area of four flower beds = 4 × 16 = 64 sq m
Remaining area = 120 − 64 = 56 sq m
Width = Area ÷ Length = 300 ÷ 25 = 12 m
Area = 500 × 200 = 1,00,000 sq m
Cost = (1,00,000 ÷ 100) × 8 = 1000 × 8 = ₹8,000
6.3 Area of a Triangle
🔍 Discovery Activity
Draw a rectangle and cut it along its diagonal. You get two equal triangles. Each triangle has exactly half the area of the rectangle!
When you draw a diagonal in a rectangle, you get two triangles that are identical — they have the same shape, same size, and same area. Each triangle = ½ of the rectangle.
📐 The Formula
For any triangle drawn inside a rectangle where the triangle’s base = rectangle’s length and triangle’s height = rectangle’s breadth:
🔢 Why does this work?
→ Area of △BAD = ½ × base × height
Triangle ABE (red) = half of (rectangle AFED + rectangle BFEC)
= half of rectangle ABCD
→ Area of △ABE = ½ × base × height
Both triangles have the SAME area but look different!
The area of any triangle = ½ × base × height. This works for all triangles — even ones that don’t look like “half of a rectangle” at first glance!
The height of a triangle is the perpendicular distance from the base to the opposite vertex — it is NOT always a side of the triangle!
Estimating Area Using Graph Paper
For irregular shapes, we can estimate area using graph/squared paper. Place the shape on the paper and count squares.
📋 Four Counting Rules
- Full square inside the shape → count as 1 sq unit
- Less than half square → ignore (count as 0)
- More than half square → count as 1 sq unit
- Exactly half square → count as ½ sq unit
🔷 Splitting Figures into Rectangles
Complex L-shaped or irregular figures can be broken into smaller rectangles or triangles. Find the area of each part and add them all up.
Draw extra lines to split the irregular shape into rectangles. Calculate each rectangle’s area separately, then add. You can also find the area of a large rectangle and subtract the missing parts.
🧩 Tangram Exploration
A tangram is a puzzle made of 7 pieces (shapes A, B, C, D, E, F, G). Key relationships:
- Shapes A and B have the same area
- Shapes C and E have the same area
- Shape D = Shapes C + E combined (Shape D is twice shape C)
- All 7 pieces together form a big square
- Same 7 pieces can also form a rectangle with the same area but different perimeter
Relationship Between Perimeter and Area
🔑 Key Concepts
- Two figures with the same perimeter can have different areas
- Two figures with the same area can have different perimeters
- Perimeter and area are independent of each other
Using 9 unit squares (area = 9 sq units), you can make figures with different perimeters — from 12 units (compact 3×3 square) up to 20 units (spread out shape). Same area, different perimeter!
📐 Adding Squares Changes Perimeter
When you attach a new square to an existing figure:
- If the square shares 1 side with the figure → perimeter increases by 2
- If the square shares 2 sides with the figure → perimeter stays the same
- If the square shares 3 sides with the figure → perimeter decreases by 2
For a fixed area (e.g., 24 sq units): the rectangle that is most spread out (like 1 × 24) has the LARGEST perimeter, while the one closest to a square has the SMALLEST perimeter.
Do NOT assume that a larger perimeter means a larger area, or vice versa. A shape with a bigger perimeter can actually have a smaller area than another shape!
Quick Summary — All Formulas at a Glance
P = 2 × (l + b)
Twice the sum of length and breadth
P = 4 × s
Four times the side length
P = a + b + c
Sum of all three sides
P = n × s
Number of sides × one side
A = l × b
Length times breadth
A = s × s = s²
Side times side
A = ½ × b × h
Half of base × height
Same area ≠ same perimeter
They are independent!
🗂️ Units Summary
cm, m, km
(linear units — just length)
sq cm (cm²), sq m (m²)
(always SQUARE units)
Important Exam Questions with Solutions
🔷 Perimeter Problems
Cost = 540 × 40 = ₹21,600
Total rope = 3 × 780 = 2,340 m
🟩 Area Problems
Number of trees = 5000 ÷ 25 = 200 trees
Area of 4 beds = 4 × (2 × 1) = 8 sq m
Area for lawn = 180 − 8 = 172 sq m
Let square have side s. Perimeter of square = 4s.
Each rectangle = s × s/2. Each rectangle’s perimeter = 2(s + s/2) = 3s.
Two rectangles combined = 3s + 3s = 6s = 1½ × 4s ✓
🔺 Area Maze Puzzle Solutions
13 : 26 = 1 : 2, so 15 : ? = 1 : 2 → ? = 30 sq cm
📚 CBSE Class 6 Mathematics | Chapter 6 — Perimeter and Area | Based on NCERT Ganita Prakash
Study well, practice daily, and you’ll master Perimeter and Area! 🌟
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