6. Measuring Space: Perimeter and Area

13 topics2 h

Chapter 6 of NCERT Mathematics for Class 9: Measuring Space: Perimeter and Area. Five sections — Perimeter, and the constant hidden in every circle, Arcs, and problems built on them, Area of straight-sided figures, Squaring a shape and Area of a circle and its parts. 13 videos, 2 h in all.

Worked answers to the 56 exercise questions in this chapter →

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Part 1

Perimeter, and the constant hidden in every circle

10 minPerimeter as a walk around the border, and why perimeter-to-side ratios are fixedRunners on a bend start on a staircase, not a line. The reason is the only thing perimeter has ever been: a walk right round the border.10 min · 12 partsहिंदी10 minWhy C/D is the same number for every circleMeasure round any circle, measure across it, divide — the same number every time. Nothing about a circle promises that, so it needs a reason.10 min · 12 partsहिंदी10 minCornering π: from inscribed polygons to Mādhava's exact seriesA six-sided figure inside a circle proves in under a minute that π is bigger than 3. Four thousand years of progress came from that one move.10 min · 12 partsहिंदी10 minπ is irrational, and what that rules outLong-divide 1 by 7 and watch the remainders, not the digits. The moment one comes back, everything after it is already decided.10 min · 12 partsहिंदी

Part 2

Arcs, and problems built on them

10 minArc length as the central angle's share of the circumferenceThe arc-length formula is a statement about sharing, and nobody had to guess it. Turning a circle about its centre changes nothing.10 min · 12 partsहिंदी10 minPerimeter puzzles: composite curved boundaries reduce to arcs you already knowA boundary made of curves looks like it needs something new. It does not — every one of these figures is just a list of arcs.10 min · 12 partsहिंदी

Part 3

Area of straight-sided figures

10 minFrom rectangle to parallelogram: area survives rearrangementBase times height is not a second rule to learn beside length times width. Cut a triangle off one end and carry it to the other.10 min · 12 partsहिंदी10 minA triangle is half a parallelogramHalf base times height is not really a fact about triangles. It is the discovery that area is blind to everything else about the shape.10 min · 12 partsहिंदी10 minHeron's formula: area from the three sides aloneHeron's formula usually arrives looking like a conjuring trick. It is not: three sides pin a triangle down, so the area was already decided.10 min · 12 partsहिंदी10 minBrahmagupta's formula, and Heron's as the case where a side vanishesThree sides fix a triangle, so four sides ought to fix a quadrilateral. Four strips of card show in seconds that they do not.10 min · 12 partsहिंदी

Part 4

Squaring a shape

10 minBaudhāyana's construction: turning a rectangle into a square of matching areaSquaring a shape does not mean multiplying it by itself. It means building a square that holds exactly as much — with a compass, in six steps.10 min · 12 partsहिंदी

Part 5

Area of a circle and its parts

10 minSlicing a disc into sectors to see where πr² comes fromπr² is not a second discovery about circles. It is the circumference formula rearranged, and you can watch one turn into the other.10 min · 12 partsहिंदी10 minA sector's area is its angle's share of the wholeA sector takes its angle's share of the disc, for exactly the reason an arc takes its share of the circumference. One argument, used twice.10 min · 12 partsहिंदी

What you will be able to do

  • State what a perimeter is as a distance travelled around a border, and apply that description to a figure with no formula attached to it
  • State the claim of §6.2 precisely: the quotient C/D takes the same value for every circle, whatever its size
  • Explain why a polygon drawn inside a circle gives a lower bound for π, and one drawn outside gives an upper bound
  • State what a rational number is, in the form the chapter uses, and give examples in each of the three shapes the chapter offers
  • Convert between the two forms of the circumference formula and say why both are in use
  • Decompose a composite curved boundary into arcs, stating each arc's radius and its fraction of a full circle
  • State what a unit of area is and why area is always reported relative to one
  • Derive the area of a right-angled triangle as half the enclosing rectangle
  • Explain why a side-only formula for a triangle's area must exist, from the rigidity of a triangle
  • Demonstrate that four given side lengths do not determine a quadrilateral's area, by hinging or from the chapter's own three figures
  • State what "squaring a shape" asked for in ancient practice, and what the target is for a rectangle of sides a and b
  • Explain why the ratio of the square of a perimeter to an area is fixed for a family of scale copies, and compute it for a square and for an equilateral triangle

What this chapter assumes you already know

  • Length as an additive quantity: the length of a path split into pieces is the sum of the pieces
  • Perimeter formulas for square, rectangle and triangle from Class 8, at the level of "I can use them"
  • Ratio written as m : n, and the fact that multiplying both terms by the same number leaves the ratio unchanged
  • Radius and diameter of a circle as named parts of the figure
  • That a scale copy of a figure multiplies every length in it by one common factor
  • Circumference, diameter and radius as named parts of a circle, and D = 2r
  • Ratio, and the fact that a common factor cancels from both terms
  • Rounding a measurement, and the idea that a measured value carries an error

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 means the formula for the shape's perimeter."
  • "4a is something to remember."
  • "Doubling the side doubles the ratio."
  • "The circle's perimeter divided by its radius is the standard ratio."
  • "The outside runner has further to go, so the stagger is unfair to her."
  • "A ratio that holds for two examples holds in general."
  • "People measured a lot of circles and the answers agreed, so the ratio is constant."
  • "The ratio is constant because π is constant."