6. Perimeter and Area

8 topics1 h

Chapter 6 of NCERT Mathematics for Class 6: Perimeter and Area. Four sections — Perimeter, Area, Area and perimeter are independent and Area of a triangle. Eight videos, 1 h in all.

Worked answers to the 30 exercise questions in this chapter →

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Part 1

Perimeter

10 minPerimeter as the distance all the way roundPerimeter is the length of a walk round the edge — so a boundary can only be counted when every step is the same length. On a dot grid they are not, and the book's own exercise proves it: two of its figures are both ten steps round, and one is fifteen per cent longer than the other.10 min · 12 partsहिंदी10 minThe rectangle and square formulas are shortcuts for the same additionBoth perimeter formulas are the same addition with the shape's own repetitions gathered up — which is why they run backwards, and why they stop dead the moment the repetition does. Cut a rectangle in half and rejoin the pieces four ways and one rule explains every answer: the boundary is 32 minus twice the length the pieces touch along.10 min · 12 partsहिंदी10 minTriangles and regular polygons: when equal sides let you multiplyThe multiplication shortcut needs equal sides and nothing else — but “regular” demands equal angles too, which is more than the shortcut ever uses. Two shapes settle it: a pushed-over four-sided figure with wrong angles still gives 4 × side, and a rectangle with perfect angles does not.10 min · 12 partsहिंदी

Part 2

Area

10 minArea as a count of unit squaresArea is a count before it is ever a multiplication — length × breadth is just a fast way to count rows of unit squares, which is exactly why it fits rectangles and nothing else. The tangram proves the point with no ruler in the room: seven pieces, no measurement printed anywhere, and every area settled by laying one shape on another.10 min · 12 partsहिंदी10 minWhy area is measured in squares and not in circlesA unit of area has to give the same answer twice, and circles fail that test — not because they are curved, but because they leave gaps, so the count depends on how you packed them. The book's own pair of pictures shows the damage: one rectangle, two honest counts, 42 and 44.10 min · 12 partsहिंदी9 minEstimating the area of a shape with no formulaThe four counting rules on this page are a deliberate trade: give up exactness at the boundary and get back a procedure that returns a number for any shape at all. Two of the rules are wrong on purpose — and because they are wrong in opposite directions, the errors pay each other back all the way round the outline.9 min · 12 partsहिंदी

Part 3

Area and perimeter are independent

11 minSame area, many perimeters: why one does not determine the otherPin the area and the boundary still swings wildly — and the mechanism is one line: attaching a square changes the perimeter by four minus twice the edges the join hides, with the area you added appearing nowhere in it. Every hole-free nine-square figure was enumerated to settle the book's open question: 1248 of them, and only one reaches the smallest perimeter.11 min · 12 partsहिंदी

Part 4

Area of a triangle

9 minWhy a triangle takes exactly half the rectangle around itA triangle takes exactly half the rectangle around it — whatever shape the triangle is. That needs an argument, not a formula.9 min · 12 partsहिंदी

What you will be able to do

  • State what perimeter measures and why it is a length, not a region
  • Derive the rectangle rule by adding the four sides and collecting the repeats
  • Compute the perimeter of a triangle from three given side lengths
  • State what area measures and distinguish it from perimeter
  • State what a shape must do to serve as a unit of area
  • Explain why some closed shapes have no area rule and still have an area
  • State that two figures may share an area and differ in perimeter, and the other way round
  • Cut a rectangle along a diagonal and check that the two triangles coincide

What this chapter assumes you already know

  • Measuring a length in centimetres and metres with a scale or measuring tape
  • Multiplying and dividing whole numbers, and adding lengths in the same unit
  • That a rectangle's opposite sides are equal and a square's four sides are all equal
  • Multiplication as repeated addition, and taking a common factor outside a bracket
  • Multiplication of whole numbers, and division as its inverse
  • Reading measurements in metres and centimetres from a diagram
  • Earlier classes: working out how much a rectangle or a square covers, using squared paper — the recall this section opens by asking for
  • Drawing a circle of a given diameter, and reading a graph sheet

It builds directly on A point fixes a location; a segment is the shortest route between two points, Shapes come in sequences too, with rules of their own, Acute, obtuse and reflex: one classification covering every angle and Common factors: which jump sizes can land on a number.

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "Perimeter is how much space is inside."
  • "The longer track means the longer run."
  • "Three rounds of rope needs three times the length of the field."
  • "Nine steps means nine units."
  • "So the diagonal is about the same, call it 9."
  • "6s + 3d should be simplified to 9 units."
  • "Only shapes with a formula have a perimeter."
  • "Perimeter of a rectangle is length × breadth."