Complete Summary, Explanations, and Solutions for Perimeter and Area – Ganita Prakash Class VI, Chapter 6 – Rectangles, Squares, Triangles, Questions, Answers

Chapter at a Glance – Perimeter and Area

This chapter covers the concepts of perimeter as the boundary length and area as the enclosed region for plane figures, with formulas, calculations, and activities.

Main Topics Covered

Perimeter: Definition, formulas for rectangle, square, triangle, regular polygons
Examples and real-life applications (fencing, lace)
Activities: Running tracks, estimate perimeters, diagonals vs straight
Area: Definition, formulas for rectangle, square, triangle
Estimating areas on grids, why squares for measurement
Same area different perimeters, house plans, mazes

Key Takeaways for Exams

Perimeter Formulas

Rectangle: 2(l+b), Square: 4s, Triangle: sum sides.

Area Formulas

Rectangle: l×b, Square: s², Triangle: half rectangle.

Regular Polygons

Perimeter: n×side.

Same Area Perimeters

Different shapes same area, vary perimeters.

Estimations

Grid counting, ignore half.

Applications

Fencing cost, uncarpeted area, trees in grove.

Key Formulas & Properties – Perimeter and Area

Essential formulas for calculations.

Perimeter Formulas

Shape	Formula	Example
Rectangle	2×(l + b)	2×(12+8)=40 cm
Square	4×side	4×1=4 m
Triangle	Sum of sides	4+5+7=16 cm
Equilateral Triangle	3×side	3×AB
Regular Polygon	n×side	5×side for pentagon

Area Formulas

Shape	Formula	Example
Rectangle	l × b	5×4=20 sq m
Square	side²	3²=9 sq m
Triangle	Half rectangle	Half of base×height

Properties

Same Area Different Perimeters: Possible, e.g., rectangles 24 sq units.
Diagonals Longer: Perimeter with diagonals > straight.
Estimation Rules: Count full, >half as 1,

Concept Cards – Quick Explanations

Perimeter

Boundary length of closed figure.

Area

Enclosed region measure.

Rectangle Perimeter

2(l + b)

Square Perimeter

4×side

Triangle Perimeter

Sum sides

Regular Polygon

Equal sides/angles, perimeter n×side.

Rectangle Area

l × b

Square Area

side²

Triangle Area

Half rectangle.

Grid Estimation

Count squares with rules.

Examples + Solutions

Example 1: Rectangular Tablecloth Lace

3 m long, 2 m wide. Perimeter 2(3+2)=10 m.

Example 2: Square Park Rounds

Side 75 m, perimeter 300 m, 3 rounds 900 m.

Example 3: Uncarpeted Floor

Floor 5×4=20 sq m, carpet 3×3=9 sq m, uncarpeted 11 sq m.

Example 4: Remaining Land After Flower Beds

Land 12×10=120 sq m, beds 4×16=64 sq m, remaining 56 sq m.

Example 5: Coconut Grove Trees

Grove 100×50=5000 sq m, each 25 sq m, max 200 trees.

Example 6: Tiling Cost

Plot 500×200=100000 sq m, ₹8/100 sq m, cost ₹8000.

Figure it Out Solutions (All Solved)

Section 6.1: Page 132

1a. Rectangle P=14 cm, b=2 cm, l=?

l=5 cm

1b. Square P=20 cm, side=?

side=5 cm

1c. Rectangle P=12 m, l=3 m, b=?

b=3 m

2. Wire rectangle to square side?

4 cm

3. Triangle third side, P=55 cm, sides 20,14 cm

21 cm

4. Fencing cost 150×120 m, ₹40/m

₹21,600

5a. String 36 cm square side

9 cm

5b. Equilateral triangle side

12 cm

5c. Hexagon side

6 cm

6. Field fencing 3 rounds, 230×160 m

2340 m

Section 6.1: Page 133

1. Akshi 5 rounds distance

1100 m

2. Toshi 7 rounds, who longer?

1260 m, Toshi

Deep Dive: Page 134

Starting points A, B for 350 m

A: 50 m before flag on side, B: 25 m before on outer.

Page 134: Triangle Perimeter

Diagonal perimeter >9?

Yes, diagonals longer than sides.

Page 135: Perimeters in s+d

Figures perimeters

8s+2d, 4s+6d, 12s+6d, 18s+6d

Regular Polygons

General perimeter

n × side

Section 6.2: Page 138

1. Garden width, l=25 m, A=300 sq m

12 m

2. Tiling cost 500×200 m, ₹8/100 sq m

₹8000

3. Coconut trees max, 100×50 m, 25 sq m each

200

4a. Split figure area

28 sq m

4b. Split figure area

9 sq m

Page 140: Areas

Figures areas

4,9,10,11 sq units

Section 6.3: Page 144

1a-e. Areas by dividing

a.24, b.30, c.48, d.16, e.12 sq units

Page 145: 9 Squares

1. Smallest perimeter

12 units (3x3 square)

2. Largest perimeter

20 units (1x9 rectangle)

3. 18 units figure

Possible shapes

4. Multiple shapes?

Yes except smallest.

Page 146: Charan House

a. Missing measurements

Small bed 15×12=180, Utility 15×3=45, Hall 20×12=240, Parking 15×3=45, Garden 20×3=60 sq ft

b. Area

1050 sq ft

Page 147: Sharan House

a. Missing

Utility 7×10=70, Hall 23×15=345, Entrance 7×15=105, Small bed 12×10=120, Toilet 5×10=50 sq ft

b. Area

1050 sq ft, same area, Sharan perimeter 134 > Charan 130 ft

Page 148: Mazes

a-d. Missing values

a.30, b.9, c.16 sq cm, d.5 cm

Page 149: Figure it Out

1. Sum areas rectangle dimensions

16×4 or 8×8 etc.

2. Garden width

20 m

3. Uncarpeted

11 sq m

4. Lawn area after beds

172 sq m

5. Shapes A=18, B=20 sq units, A longer P

E.g., A 18×1 P=38, B 5×4 P=18

8. Square folded cut statement

c. Perimeters added 1.5 times square

Extra Practice Questions (Exam-Ready) – Chapter 6

25+ Questions • Categorized by Marks • With Detailed Solutions • Difficulty Tags

1-Mark Questions (Very Short Answer)

1. Perimeter rectangle 10×6 cm?

2. Square side if P=36 cm?

3. Area square 7 cm side?

4. Equilateral triangle P=24 cm, side?

5. Uncarpeted if floor 8×5, carpet 4×3?

2-Mark Questions (Short Answer)

6. Fencing cost 100×80 m, ₹50/m?

7. Third side triangle P=30 cm, sides 9,11 cm?

8. Area rectangle l=15 m, b=8 m?

9. Max trees 200×100 m, 20 sq m each?

10. Hexagon side if P=42 cm?

3-Mark Questions (Reasoning)

11. Why diagonals increase perimeter?

12. Same area min/max perimeter rectangles 36 sq units?

13. Why squares for area measurement?

14. Triangle area relation to rectangle?

15. Wire 40 cm square side?

4–5 Mark Questions (Application)

16. Remaining area land 20×15 m, 4 beds 2×2 m?

17. Tiling cost 300×200 m, ₹10/100 sq m?

18. House area from plan missing dims.

19. 3 rounds rectangle track 100×60 m?

20. Split figure area 2 rectangles.

Challenge Questions (6+ Marks)

21. Explain same perimeter different areas.

22. Grid estimate irregular shape area.

23. Fencing 4 rounds field 250×180 m.

24. Uncarpeted with multiple carpets.

25. Regular octagon side 5 cm perimeter.

Common Mistakes & How to Avoid

Mistake 1: Confusing Perimeter/Area

Using area formula for perimeter.

Avoid: Remember perimeter boundary, area inside.

Mistake 2: Forgetting ×2 in Rectangle P

Just l+b.

Avoid: Think opposite sides.

Mistake 3: Wrong Units

cm for area, sq cm not.

Avoid: Square for area.

Mistake 4: Grid Count Errors

Ignoring half rules.

Avoid: Follow conventions.

Mistake 5: Missing Rounds in Fencing

Single perimeter for multiple.

Avoid: Multiply by rounds.

Mistake 6: Triangle Area Wrong

Not half.

Avoid: Diagonal halves rectangle.

History & Fun Facts

Ancient Origins

Egyptians measured land areas for taxation ~3000 BC.

Greeks formalized perimeter (peri around, metron measure).

Real-Life Applications

Agriculture: Fencing fields, planting density.
Construction: Flooring costs, borders.
Sports: Track perimeters for distances.
Design: Fabrics, wallpapers.

Fun Facts

Square maximizes area for given perimeter.
Pi from circle perimeter/area relations.
Tangrams from China, rearrange same area.
Fractals like Koch infinite perimeter finite area.
Honeybees use hexagons for efficiency.

Did You Know?

Archimedes approximated pi using perimeters.

Quick Revision One-Pager

Formulas

Type	Formula
Rectangle P	2(l+b)
Square P	4s
Triangle P	a+b+c
Rectangle A	l×b
Square A	s²

Quick Rules

✓ Perimeter: Sum boundaries.
✓ Area: Count enclosed.
✓ Triangle: Half base×height.
✓ Same A different P: Elongate increases P.
✓ Grid: Full=1, >0.5=1, =0.5=0.5, <0.5=0.

Mind Map

Central: Perimeter & Area

Perimeter: Formulas, activities, regular shapes
Area:
- Formulas, estimation
- Triangle relation, same A diff P
Applications: Cost, remaining

Exam Tips

Before Solving

Identify shape/formula

During Solving

Units consistent

After Solving

Check reasonable

Time-Savers

Formulas memorize

Interactive Quiz – 15 Questions

Test Your Perimeter & Area Knowledge!