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Perimeter and Area – Class 6 Maths Chapter 6 Notes & NCERT PDF

Class 6MathematicsChapter 6NCERT book: Ganita Prakash - Mathematics Class 6

Notes and a simple summary of Chapter 6 of the NCERT Class 6 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 6: Perimeter and Area

Introduction

Chapter 6 of Ganita Prakash revisits two ideas from earlier grades and takes them further:

Perimeter is the distance covered along the boundary of a closed figure when you go around it once.
Area is the amount of region enclosed by a closed figure.

Children like Debojeet, Akshi, Toshi, Usha, Charan and Sharan meet these ideas in real situations: taping a photo frame, putting lace on a tablecloth, running around tracks and planning a house.

Key Ideas

6.1 Perimeter

For any polygon, perimeter = the sum of the lengths of all its sides.

  • Rectangle: perimeter = length + breadth + length + breadth = 2 × (length + breadth). Rectangle ABCD, 12 cm by 8 cm: 2 × (12 + 8) = 40 cm.
  • Square: perimeter = 4 × side. Debojeet needs 4 × 1 m = 4 m of tape for a square frame of side 1 m.
  • Triangle: add the three sides. A triangle with sides 4 cm, 5 cm and 7 cm has perimeter 16 cm.
  • Equilateral triangle: perimeter = 3 × side. More generally, for a regular polygon (all sides and all angles equal), perimeter = number of sides × side.

    • Matha Pachchi (running tracks): Akshi runs 5 rounds of the outer track (70 m × 40 m), and Toshi runs 7 rounds of the inner track (60 m × 30 m).

  • One outer round = 2 × (70 + 40) = 220 m, so Akshi runs 5 × 220 = 1100 m.
  • One inner round = 2 × (60 + 30) = 180 m, so Toshi runs 7 × 180 = 1260 m.
  • Toshi runs farther, even though her track is smaller.

    • Deep Dive: runners on different-sized square tracks need different starting points to share one finish line.

    Straight and diagonal units: on dot paper, diagonal steps are longer than straight steps. Akshi says the triangle's perimeter is 9 units; Toshi says it is more. Toshi is right: it is written 6s + 3d.

    Split and rejoin: a 6 cm × 4 cm chit is cut into two 6 cm × 2 cm pieces and joined in different ways. The area stays the same, but the perimeter changes. Arrangement (a), end to end, has perimeter 28 cm.

    6.2 Area

  • Area of a rectangle = length × breadth. Area of a square = side × side.
  • Area is measured in square units (sq cm, sq m).

    • Estimating on squared paper:

  • A full square counts as 1 sq unit.
  • Ignore parts that are less than half a square.
  • Count parts that are more than half a square as 1.
  • Count an exactly half square as ½.

    • Why squares? Circles cannot cover a region without gaps. The same rectangle can hold 42 circles one way and 44 another, so circles give unreliable areas. Squares tile a region with no gaps or overlaps.

    Tangram:

  • Shapes A and B have equal area, and shapes C and E have equal area.
  • Shape D is exactly covered by C and E, so D has twice the area of C.
  • The same 7 pieces can form a square or a rectangle: the area is the same, but the perimeter can differ.

    • Same area, different perimeters: rectangles with whole-number sides and area 24 sq units:

  • 1 × 24 → perimeter 50
  • 2 × 12 → perimeter 28
  • 3 × 8 → perimeter 22
  • 4 × 6 → perimeter 20

    • The long, thin rectangle has the greatest perimeter, and the most square-like one has the least.

    6.3 Area of a triangle

    Cut a rectangle along a diagonal and you get two triangles that match exactly. So:

    The area of a triangle drawn inside a rectangle on the same base and with the same height is half the area of the rectangle.

    For triangle ABE, whose top corner E is not at a corner of the rectangle, split it with the line EF. Each part is half of a smaller rectangle, so the whole triangle is still half of ABCD.

    Making it 'More' or 'Less'

    With 9 unit squares:

  • the smallest perimeter is 12 (a 3 × 3 square);
  • the largest is 20 (all 9 in a line).

    • Moving one square around can make the perimeter go up, go down or stay the same. The chapter also has house plans (Charan's and Sharan's), where you find missing room sizes and areas, and Area Maze puzzles.

    Worked Examples

  • Lace for a tablecloth, 3 m × 2 m (Akshi). Perimeter = 2 × (3 + 2) = 10 m of lace.
  • Usha walks 3 rounds of a square park of side 75 m. One round = 4 × 75 = 300 m, so she walks 3 × 300 = 900 m.
  • Floor 5 m × 4 m with a 3 m × 3 m carpet. Floor area 20 sq m minus carpet 9 sq m = 11 sq m not carpeted.
  • Land 12 m × 10 m with four square flower beds of side 4 m in the corners. 120 − 4 × 16 = 56 sq m left.
  • Fencing a park 150 m × 120 m at ₹40 per metre. Perimeter = 2 × (150 + 120) = 540 m, so the cost is 540 × 40 = ₹21,600.
  • A wire rectangle 5 cm × 3 cm is bent into a square. The wire length is 2 × (5 + 3) = 16 cm, so each side of the square is 16 ÷ 4 = 4 cm.

    • Figure it Out: Hints

  • Missing sides: perimeter 14 cm and breadth 2 cm means length + breadth = 7, so the length is 5 cm.
  • A 36 cm string makes a square of side 9 cm, an equilateral triangle of side 12 cm, or a regular hexagon of side 6 cm.
  • The farmer's field, 230 m × 160 m with 3 rounds of rope, needs 3 × 780 = 2340 m.
  • Coconut grove 100 m × 50 m with each tree needing 25 sq m: 5000 ÷ 25 = 200 trees.
  • Tiling 500 m × 200 m at ₹8 per hundred sq m: 100,000 ÷ 100 × 8 = ₹8000.
  • Folded square: when a square is cut in half, the two rectangles together have 1½ times the square's perimeter. You gain two new edges, each the length of a side.

    • Maths Words

  • Perimeter (परिमाप)
  • Area (क्षेत्रफल)
  • Polygon (बहुभुज)
  • Regular polygon (सम बहुभुज)
  • Rectangle (आयत)
  • Square (वर्ग)
  • Equilateral triangle (समबाहु त्रिभुज)
  • Diagonal (विकर्ण)
  • Square unit (वर्ग इकाई)
  • Boundary (सीमा)
  • Breadth / width (चौड़ाई)
  • Tangram (टैनग्राम)
  • Grid paper (वर्गांकित कागज़)

    • Common Mistakes

  • Writing area in plain units (m) instead of square units (sq m), or perimeter in square units.
  • Adding only two sides of a rectangle. Perimeter is 2 × (l + b), not l + b.
  • Assuming a bigger area always means a bigger perimeter. Figures can have the same area and different perimeters, and the reverse.
  • Counting diagonal steps on dot paper as the same length as straight steps.
  • Forgetting to change the rate unit, as in "₹8 per hundred sq m".

    • Maths in Daily Life

  • Perimeter: fencing fields, framing photos, putting lace on cloth, measuring a running track.
  • Area: buying tiles, carpets and paint, planning rooms, and deciding how many trees or crops fit on a piece of land.

    • 💡 Key Learning Points

    • ✓Find perimeters of rectangles (2 × (l + b)), squares (4 × side), triangles and regular polygons
    • ✓Find areas of rectangles and squares, and estimate irregular areas on squared paper
    • ✓Understand that a triangle drawn in a rectangle on the same base and height has half the rectangle's area
    • ✓See that the same area can have different perimeters, and the same perimeter different areas
    • ✓Solve real-life problems on fencing, tiling, carpeting, tracks and house plans

    👨‍🏫 Teaching Tips

    • →Take the class outdoors to walk around the playground and measure its perimeter with a measuring tape.
    • →Recreate Matha Pachchi with chalk tracks and let two students run different numbers of rounds, then compute who ran farther.
    • →Give each pair a 6 cm × 4 cm paper strip to cut in half and rejoin, recording how the perimeter changes while the area stays 24 sq cm.
    • →Use the tangram pieces at the end of the textbook: have students overlap shapes to compare areas before any calculation.
    • →On grid paper, ask students to draw every whole-number rectangle of area 24 and tabulate the perimeters to spot the pattern.
    • →Cut a paper rectangle along its diagonal to show that the two triangles match exactly.
    • →Use 9 square tiles (or cut-out squares) to build shapes with perimeters 12, 18 and 20.
    • →Draw Charan's house plan on the board and let groups fill in the missing dimensions and areas.

    📋 Assessment Questions

    1. Can the student find a missing side from a given perimeter (e.g., perimeter 14 cm, breadth 2 cm, so length 5 cm)?
    2. Can the student solve a cost problem such as fencing a 150 m × 120 m park at ₹40 per metre?
    3. Can the student explain why a triangle in a rectangle on the same base and height has half the rectangle's area?
    4. Can the student give two rectangles with the same area but different perimeters?
    5. Does the student write correct units (m for perimeter, sq m for area)?

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    Keep learning

    All chapters in Ganita Prakash - Mathematics Class 6

    1. 1. Patterns in Mathematics
    2. 2. Lines and Angles
    3. 3. Number Play
    4. 4. Data Handling and Presentation
    5. 5. Prime Time
    6. 6. Perimeter and Area
    7. 7. Fractions
    8. 8. Playing with Constructions
    9. 9. Symmetry
    10. 10. The Other Side of Zero