Why area is measured in squares and not in circles

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What to assume they know

Area as a count of unit squares: area as a count of unit squares, and the units sq m, sq cm and sq unit
Estimating the area of a shape with no formula: the four counting conventions for partial squares, which this experiment puts under strain
Drawing a circle of a given diameter, and reading a graph sheet
That the same region can be cut up in more than one way

What they should be able to do

State what a shape must do to serve as a unit of area
Explain why gaps between units make a count meaningless rather than merely inaccurate
Read the book's two circle packings and say why both counts are correct
Estimate the area of a circular region by counting grid squares
Test whether a proposed unit shape covers a region without gaps or overlaps
Distinguish the shapes that fail the covering test from those that pass it but are still not chosen
Give at least two reasons why the square is the convenient unit, and mark them as reasons rather than as the book's printed answer
Use the standard area units correctly in speech and writing

Where it usually goes wrong

"Circles are bad units because they are curved." Curvature is not the problem. The problem is that circles cannot be laid side by side without leaving space between them, and that is a fact about how they fit, not about how they look.
"You could count the gaps as well and get the right answer." Then the thing being counted is no longer the circle, and the unit has quietly become something else. Say this out loud — students propose it immediately.
"One of the two counts, 42 or 44, must be a mistake." Both are correct counts of their own packing. That is precisely the trouble: the answer depends on the arrangement, so it is not a property of the rectangle.
"Squares are used because they are easy to draw." Convenience of drawing is not the book's argument. The argument is about covering.
"Any shape that covers without gaps is as good as a square." Triangles and rectangles both cover without gaps, and the book still asks what makes the square better. Leaving that gap open honestly is better than closing it with an invented reason.
"Estimating by counting squares means the answer is wrong." It means the answer is approximate and improvable — a smaller grid gives a closer estimate. That is a different thing from being undefined, which is what the circle count is.
"Area units are just names, like cm and m." A square unit is manufactured from a length unit. Making that visible is what turns the convention into a reason.

Questions to check understanding

Say what conditions a shape must satisfy to be used as a unit of area
Explain, from a drawing, why a proposed unit leaves the count undetermined
Estimate the area of a curved region by counting grid squares
Decide which of a set of shapes will cover a region without gaps
Give reasons for the square being the standard unit
Convert between counting on a coarse grid and counting on a finer one
Measure a real floor or ground area and report it in square metres
The answer key appended to the cached PDF lists this page at the head of its footer page 4 and gives no answers for it

Examples worth working on the board

The question and the speech bubble (p.141). Under the heading "Let's Explore!" the book asks outright why squares are the usual unit, and a girl in the margin asks why circles could not be used instead. The section is one of very few places in the whole book where the reason behind a convention is raised at all.
The circle experiment (p.141). On a graph sheet, draw one circle whose diameter — the book adds the word breadth in brackets — is 3, then count squares to arrive at an estimate for the round region. No answer is printed, and the key appended to the cached PDF lists this page without giving one. For an added check only, and not as anything the book says: the exact area of such a region is a little over 7 square units.
The two packings (p.141). Two copies of one rectangle are printed side by side, each filled with circles of the same size but arranged differently. The book states that the first holds 42 and the second 44. Read off the printed page, the left packing is a plain grid of rows and the right packing is offset so that each row nests into the hollows of the one above. This is the single most important figure in the topic: same region, same unit, two different counts.
The follow-up task (p.141). Try filling the same space with triangles and with rectangles, without leaving gaps and without overlapping, then list what makes the square the best choice. The book asks for the list and prints none. Whatever the explanation offers here must be flagged as reasoning, not as the book's answer.
Two reasons worth offering, marked as not in the book. First, a square is the only one of the three whose two measurements are the same number, so one length fixes the unit completely and the grid it makes is the same in both directions. Second, a square unit is built directly on the unit of length — a square of side 1 cm is 1 sq cm — which is why the printed units in this chapter read sq m, sq cm and sq unit, and why a rectangle's area comes out as one length multiplied by another at all.
Where the answer is used (pp.137–140, 150). Every worked area in the chapter is in squares, and the Summary on p.150 says plainly that areas are normally counted in squares. Section 9 should connect the convention back to the arithmetic the reader has already been doing.
The two applied questions on p.141. Find, in square metres, the floor area outside the corridor; find, in square metres, the area of your own school playground. Both are real measuring tasks with no printed answer.

Figures to have open

The two circle packings from p.141, drawn as the book draws them — one plain grid, one offset — with the counts left for the figure to make. This is the topic's central figure and the argument does not survive without it. Redraw rather than reproduce.
A circle of diameter 3 on a unit grid, countable.
A test bench of candidate unit shapes covering one region: square, triangle, rectangle, circle, with gaps and overlaps made visible. Standard schematic; the book asks for this experiment rather than drawing it.
A 1 cm segment becoming a 1 sq cm square, for section 9. Standard schematic.
No photograph or textbook data table is needed.

Where this sits in the book

NCERT Ganita Prakash, Class 6, Chapter 6 "Perimeter and Area", §6.2 "Area", p.141 — the first "Let's Explore!" box, the circle experiment, the two packings, the follow-up task and the two applied questions
§6.2, p.140, for the counting conventions the circle experiment relies on, and for the square unit itself
Chapter Summary, p.150, for the book's closing statement about the units areas are normally counted in
Companion topics Area as a count of unit squares (area as a count) and Estimating the area of a shape with no formula (counting when the boundary crosses squares)
Solutions appendix bound into the cached PDF after the book's last printed page, footer page 4