Class 6 Perimeter and Area
Discover the mathematics of 1D boundary lengths and 2D surface coverage! Master perimeter formulas for rectangles, squares, triangles, and regular polygons, straight vs diagonal units, grid-based area estimation, triangle areas, Tangram puzzles, and the fascinating truth that shapes with identical areas can have completely different perimeters.
What are Perimeter and Area in Class 6?
Perimeter is the total linear distance around the outer boundary of a closed 2D plane figure. Area is the measure of the 2D surface region enclosed inside that boundary.
Perimeter: P = 2 × (length + breadth)
Area: A = length × breadth
Perimeter: P = 4 × side
Area: A = side × side (s²)
Perimeter: P = a + b + c
Area: A = ½ × base × height
Interactive Chapter Roadmap & Table of Contents
Perimeter – Boundary Lengths of Closed Plane Figures
The perimeter of any closed plane figure is the total distance covered along its boundary when you go around it once. For any polygon made up of straight line segments:
Perimeter of a Rectangle
Opposite sides are equal: AB = CD = l, and BC = DA = b.
P = 2 × (12 + 8) = 2 × 20 = 40 cm.
Perimeter of a Square
All 4 sides are equal in length.
Tape length = 4 × 1 m = 4 m.
Perimeter of a Triangle
Sum of the 3 side lengths: P = a + b + c.
P = 4 + 5 + 7 = 16 cm.
Tablecloth Lace (Akshi)
Tablecloth length = 3 m, width = 2 m. The lace borders all 4 sides.Lace required = 2 × (3 m + 2 m) = 10 m.
Square Park Jogging (Usha)
Square park side = 75 m. 1 round = 4 × 75 m = 300 m.Total for 3 rounds = 3 × 300 m = 900 m.
Running Track Comparison: Akshi vs Toshi
Calculate track perimeters and total distance covered across multiple laps!
1 Lap Perimeter = 2 × (70 + 40) = 220 m
Total Distance (5 laps) = 1100 m
1 Lap Perimeter = 2 × (60 + 30) = 180 m
Total Distance (7 laps) = 1260 m
Straight (s) & Diagonal (d) Grid Units
On dot-grid paper, lines connecting orthogonally adjacent dots are straight units (s), while diagonal connections across a square are diagonal units (d).
Triangle Perimeter = 6s + 3d units
Because d ≈ 1.414 > s, diagonal segments are longer than straight segments!
Regular Polygon Perimeter Calculator
A regular polygon has equal sides and equal angles.Perimeter = n × (length of one side)
Split and Rejoin Activity (6 cm × 4 cm Paper Chit)
When a 6 cm × 4 cm paper chit is cut into two 6 cm × 2 cm pieces, joining them side-by-side yields a 12 cm × 2 cm strip with perimeter 28 cm. Joining them into an L-shape produces a perimeter of 22 cm.
Key Theorem: Rearranging pieces preserves the exact same area (24 sq cm) but creates different boundary perimeters!
Area – Surface Coverage and Square Units
The area of a closed figure is the amount of 2-dimensional surface region enclosed within its boundaries. Standard formulas include:
Area = length × breadth
Floor (5 m × 4 m) = 20 sq m. Carpet (3 m × 3 m) = 9 sq m.Uncarpeted floor area = 20 − 9 = 11 sq m.
Area = side × side (s²)
Land (12 m × 10 m) = 120 sq m. 4 beds of (4 m × 4 m) = 64 sq m.Remaining land area = 120 − 64 = 56 sq m.
Estimating Area Using Grid Paper (The 4 NCERT Conventions)
Count as exactly 1 sq unit.
Portions < 0.5 are ignored (0).
Portions > 0.5 count as 1 sq unit.
Count as exactly ½ (0.5) sq unit.
Tangram Area Breakdown (7 Pieces)
Let small triangle C have Area = 1 unit:
- Small Triangles C & E = 1 unit each
- Square D, Parallelogram F, Med Triangle G = 2 units each
- Large Triangles A & B = 4 units each
- Complete Tangram Square = 16 units (16 × Area of C)
Same Area, Different Perimeters (Area = 24 sq units)
Area of a Triangle and Irregular Figures
The Rectangle–Triangle Relationship
Cutting a rectangle along its diagonal produces two identical right triangles of equal area. Furthermore, for ANY triangle with base b and perpendicular height h enclosed inside a rectangle of dimensions b × h:
Triangles with the same base and height (like Triangle BAD and Triangle ABE) have identical areas even if their visual shapes slant differently.
House Plan Analysis: Charan vs Sharan (Area = 1050 sq ft)
Compare room dimensions, total area, and boundary perimeter across two distinct floor plans.
- Master Bedroom (15×15) = 225 sq ft
- Small Bedroom (15×12) = 180 sq ft
- Hall (20×12) = 240 sq ft
- Kitchen (15×12) = 180 sq ft
- Utility (15×3) = 45 sq ft • Toilet (5×10) = 50 sq ft
- Parking (15×3) = 45 sq ft • Garden (20×3) = 60 sq ft
- Master Bedroom (12×15) = 180 sq ft
- Small Bedroom (12×10) = 120 sq ft
- Hall (23×15) = 345 sq ft
- Kitchen (18×10) = 180 sq ft
- Entrance (7×15) = 105 sq ft • Utility (7×10) = 70 sq ft
- Toilet (5×10) = 50 sq ft
NCERT Area Maze Puzzles (Click to Reveal Solutions)
Solve for missing areas and dimensions using proportional reasoning!
Top row: 13 & 26 sq cm. Bottom-left: 15 sq cm. Find bottom-right?
Stepped rectangles of 10 sq cm and side 3 cm. Find top square?
Bottom 60 sq cm, Middle 42 sq cm, Left 15 cm. Find top area?
Left 38 sq cm, bottom-right 18 sq cm (5×4 step). Find top width ?
12 Comprehensive Worked Examples
Example 1: Rectangle Perimeter
What we know: Length = 15 cm, Breadth = 9 cm
Formula: P = 2 × (length + breadth)
Step 1: Sum length and breadth: 15 + 9 = 24 cm
Step 2: Multiply by 2: 2 × 24 cm = 48 cm
Answer: Perimeter = 48 cm
Example 2: Square Perimeter
What we know: Side length = 14 m
Formula: P = 4 × side
Step 1: Multiply side length by 4: 4 × 14 m
Step 2: 4 × 14 = 56 m
Answer: Perimeter = 56 m
Example 3: Triangle Perimeter & Missing Side
What we know: Perimeter = 45 cm, Side 1 = 18 cm, Side 2 = 12 cm
Formula: Side 3 = Perimeter − (Side 1 + Side 2)
Step 1: Sum known sides: 18 + 12 = 30 cm
Step 2: Subtract from perimeter: 45 − 30 = 15 cm
Answer: Third side = 15 cm
Example 4: Regular Hexagon Perimeter
What we know: Regular hexagon has 6 equal sides, Side = 8 cm
Formula: P = 6 × side
Step 1: Multiply number of sides (6) by length (8 cm)
Step 2: 6 × 8 = 48 cm
Answer: Perimeter = 48 cm
Example 5: Multiple Rounds on Track
What we know: Track side = 80 m (square), Athlete runs 4 rounds
Formula: Total Distance = Rounds × (4 × side)
Step 1: 1 lap perimeter = 4 × 80 = 320 m
Step 2: 4 laps = 4 × 320 = 1280 m
Answer: Total distance = 1,280 m
Example 6: Fencing Cost
What we know: Field 200 m × 150 m, Rate = ₹25 per metre
Formula: Cost = Perimeter × Rate
Step 1: Perimeter = 2 × (200 + 150) = 700 m
Step 2: Total Cost = 700 × ₹25 = ₹17,500
Answer: Fencing cost = ₹17,500
Example 7: Rectangle Area
What we know: Length = 24 m, Breadth = 15 m
Formula: Area = length × breadth
Step 1: Multiply 24 by 15
Step 2: 24 × 15 = 360 sq m
Answer: Area = 360 sq m
Example 8: Remaining Land Area
What we know: Land 15 m × 12 m (180 sq m), 4 beds of 2 m × 1 m (8 sq m)
Formula: Remaining = Land Area − Total Bed Area
Step 1: 4 flower beds = 4 × (2 × 1) = 8 sq m
Step 2: Remaining = 180 − 8 = 172 sq m
Answer: Available lawn area = 172 sq m
Example 9: Estimating Grid Area
What we know: Leaf shape covers 8 full squares, 4 more-than-half squares, 3 less-than-half
Formula: Estimated Area = (Full × 1) + (More-than-half × 1)
Step 1: 8 + 4 = 12 counted squares (ignore less-than-half)
Step 2: Total = 12 × 1 = 12 sq units
Answer: Estimated area = 12 sq units
Example 10: Area of a Triangle
What we know: Base = 16 cm, Height = 9 cm
Formula: Area = ½ × base × height
Step 1: ½ × 16 = 8 cm
Step 2: 8 × 9 = 72 sq cm
Answer: Area = 72 sq cm
Example 11: Irregular Rectilinear Area
What we know: L-shape split into Rectangle A (6×2 = 12) and Rectangle B (4×3 = 12)
Formula: Total Area = Area(A) + Area(B)
Step 1: Calculate sub-rectangles: 12 + 12
Step 2: 12 + 12 = 24 sq m
Answer: Total area = 24 sq m
Example 12: Same Area, Different Perimeter
What we know: Square A (6×6, Area=36) vs Rectangle B (9×4, Area=36)
Formula: P(Square) = 4s; P(Rect) = 2(l+b)
Step 1: P(A) = 4 × 6 = 24 units
Step 2: P(B) = 2 × (9 + 4) = 26 units
Answer: Both have Area = 36 sq units, but Rectangle B has a larger perimeter (26 > 24).
20 Core Practice Questions
What is the perimeter of a rectangle with length 12 cm and breadth 8 cm?
Debojeet tapes around a square photo frame of side 1 m. What total length of tape does he need?
A triangle has sides 4 cm, 5 cm, and 7 cm. What is its perimeter?
Akshi puts lace around a tablecloth 3 m long and 2 m wide. How much lace is required?
Usha runs 3 rounds around a square park of side 75 m. What total distance does she cover?
A wire rectangle of 5 cm × 3 cm is straightened and bent into a square. What is the side of the square?
A triangle has a perimeter of 55 cm. If two sides are 20 cm and 14 cm, what is the third side?
What is the cost of fencing a rectangular park 150 m long and 120 m wide at ₹40 per metre?
A piece of string 36 cm long is formed into an equilateral triangle. What is the length of each side?
A farmer fences a rectangular field (230 m × 160 m) with 3 rounds of rope. What total rope is needed?
Akshi runs 5 rounds on a 70 m × 40 m track. Toshi runs 7 rounds on a 60 m × 30 m track. Who ran further?
A room floor is 5 m × 4 m. A square carpet of side 3 m is laid. What area of the floor is NOT carpeted?
Four square flower beds (4 m side each) are in the corners of a land (12 m × 10 m). What is the remaining area?
A rectangular garden of area 300 sq m has a length of 25 m. What is the width of the garden?
What is the cost of tiling a rectangular plot (500 m × 200 m) at ₹8 per hundred sq m?
A coconut grove is 100 m × 50 m. If each tree requires 25 sq m, how many trees can be planted?
What is the area of a right-angled triangle with base 8 cm and height 6 cm?
Using 9 unit squares, what are the smallest and largest possible perimeters for a connected shape?
Charan's house is 35 ft × 30 ft (Area = 1050 sq ft). Sharan's house is 42 ft × 25 ft (Area = 1050 sq ft). Which house has the larger perimeter?
A square of side s is folded in half and cut into two rectangles. The sum of perimeters of the two rectangles is:
10 Challenge & Figure It Out Questions
The Common Finishing Line Stagger (Section 6.1 Deep Dive)
Two concentric square tracks share a common finish line in the middle of their right vertical sides. The inner track has side 100 m (perimeter 400 m) and the outer track has side 150 m (perimeter 600 m). For a 350 m race, where should runners A (inner) and B (outer) start?
Straight vs Diagonal Units on Dot Grid (Section 6.1)
Why is the diagonal distance d between adjacent diagonal dots on a 1×1 square grid strictly greater than the straight distance s? If a shape boundary has 6 straight segments and 3 diagonal segments, why can its perimeter not be exactly 9 units?
Split and Rejoin: Boundary Invariance vs Variation (Section 6.1)
A rectangular paper card 6 cm × 4 cm (Area = 24 sq cm, Perimeter = 20 cm) is cut horizontally into two 6 cm × 2 cm pieces. Show how rearranging these two pieces can produce perimeters of 28 cm and 22 cm, while keeping the area identical.
Tangram Area Hierarchy (Section 6.2)
In the standard 7-piece Tangram (A, B large triangles; C, E small triangles; D square; F parallelogram; G medium triangle), express the area of every piece and the whole big square in terms of small triangle C.
The 24 sq units Perimeter Spectrum (Section 6.2)
List all possible whole-number rectangles having an area of 24 square units. Order them from greatest to least perimeter. What geometric pattern emerges?
Equal Base & Height Triangle Equivalence (Section 6.3)
In rectangle ABCD (6 units × 4 units), triangle BAD is a right triangle formed by diagonal BD. Point E is any arbitrary point along top edge CD. Prove why Triangle ABE has the EXACT same area as Triangle BAD.
Area Maze Puzzle A (Section 6.3)
A 2×2 block of rectangles has top-left area = 13 sq cm, top-right area = 26 sq cm, and bottom-left area = 15 sq cm. Find the area of the bottom-right rectangle without measuring.
Area Maze Puzzle D (Section 6.3)
A composite shape consists of two connected rectangles: Left rectangle has area 38 sq cm and top side ? cm. Connected bottom-right rectangle has area 18 sq cm, length 5 cm, and height 4 cm below the top overlap. Find the missing top dimension ? cm.
Charan vs Sharan House Plan Cost Analysis (Section 6.3)
Charan's plot is 35 ft × 30 ft (Area 1050 sq ft) and Sharan's plot is 42 ft × 25 ft (Area 1050 sq ft). If boundary fencing costs ₹150 per foot, which homeowner pays more for fencing and by how much?
Decomposing Irregular Figures on Dot Grid (Section 6.3)
A composite polygon on a grid is bounded by vertices (0,0), (6,0), (6,4), (3,6), (0,4). Decompose this pentagon into a rectangle and a triangle to find its exact area.
Perimeter and Area – Key Takeaways
Essential definitions, formulas, and geometrical theorems to remember.
1. Perimeter Concept
Perimeter is the 1D boundary length. Rectangle P = 2(l + b), Square P = 4s, Triangle P = a + b + c.
2. Regular Polygons
For any regular polygon with n equal sides, Perimeter = n × side length.
3. Area Definition
Area measures 2D enclosed surface in square units. Rectangle A = l × b, Square A = s².
4. Triangle Area Formula
Area of any triangle = ½ × base × height, because every triangle is half of an enclosing rectangle.
5. Isoperimetric Principle
Two figures can share identical areas while having completely different perimeters.
6. Grid Decomposition
Irregular shapes are estimated via the 4 grid rules or decomposed into simpler rectangles and triangles.
16 FAQs on Perimeter and Area
Computational Thinking with Perimeter, Area & Spatial Grids
Grid-based area estimation and polygon bounding boxes form the foundational logic of computer vision rasterization, collision detection algorithms in video games, and image mask segmentation in artificial intelligence.