NCERT Solutions Ganita Prakash Chapter 6 Why squares? · Rectangles of a given area — Let’s Explore
Book page 141 & 142 Updated on2026-09-05
Q1.
Why is area generally measured using squares? Draw a circle on a graph sheet with diameter (breadth) of length 3. Count the squares and use them to estimate the area of the circular region.
Answer
Counting the circle first. Draw a circle of diameter 3 units on graph paper. Using the four counting rules you will find about
Area of the circular region ≈ 7 square units
(The exact value is about 7.1 square units, so counting squares comes very close.)
Now the main question — why squares and not circles?
Circles leave gaps. However you pack circles, small curved gaps are always left between them, so they can never measure a region exactly. In the book's picture the same rectangle holds 42 circles one way and 44 circles the other way — the answer changes with the packing, which is useless for measurement.
Squares tile perfectly. Unit squares fit together with no gaps and no overlaps, and they cover the whole region.
Squares match our length units. A square of side 1 cm gives 1 sq cm, so area and length units stay linked.
Squares are easy to count in rows and columns, which is what gives length × width.
The key phrase: a good unit of area must tile the plane — cover it completely with no gaps and no overlaps. Circles fail this test; squares pass it.
Q2.
Try using different shapes (triangle and rectangle) to fill the given space (without overlaps and gaps) and find out the merits associated with using a square shape to find the area rather than another shape. List out the points that make a square the best shape to use to measure area.
Answer
Triangles and rectangles do fill a space without gaps — so they could be used. Even so, the square wins. Here is the comparison.
Unit shape
Fills space without gaps?
Problem with it
Circle
No
Always leaves gaps; the count changes with the packing
Triangle
Yes
Must be flipped up and down alternately; hard to count in rows
Rectangle
Yes
Has two different side lengths, so the unit must be described twice
Square
Yes
No problem — the best choice
Points that make the square the best unit of area:
Squares fit together with no gaps and no overlaps.
All four sides are equal, so one number (the side) describes the unit completely.
They line up in neat rows and columns, which turns counting into a multiplication: rows × columns.
They can be cut into smaller squares (a 1 cm square = 100 squares of 1 mm), so we can measure as finely as we like.
Half a square is easy to recognise, which helps when a shape cuts across a square.
Q3.
Find the area (in square metres) of the floor outside of the corridor.
Answer
This one you measure in your own school. Here is how to do it and a sample calculation.
Take a measuring tape and measure the length and the breadth of the floor space outside the corridor, in metres.
If the space is a rectangle, area = length × breadth.
If it bends around a corner, split it into rectangles, find each area and add — exactly like the staircase figure on page 138.
Sample: length = 12 m, breadth = 3 m
Area = 12 m × 3 m = 36 sq m
Tip: if you have no tape, walk the length heel-to-toe and count steps. One ordinary step of a Class 6 student is roughly 0.5 m, so 24 steps ≈ 12 m. Measure your own step once and you will always have a “ruler” with you.
Q4.
Find the area (in square metres) occupied by your school playground.
Answer
Measure your own playground — but here is the method with a worked example.
Measure the length and the breadth of the playground in metres (use a long tape, or count steps and multiply).
Multiply: area = length × breadth.
For an L-shaped or irregular ground, split it into rectangles and add the areas.
Sample: a playground 60 m long and 40 m wide
Area = 60 m × 40 m = 2400 sq m
Perimeter (if you jog once round it) = 2 × (60 + 40) = 200 m
Did you know? A full-size cricket ground has a boundary about 65–70 m from the pitch, and a standard football field is about 100 m × 64 m = 6400 sq m.
Q5.
On a squared grid paper (1 square = 1 square unit), make as many rectangles as you can whose lengths and widths are a whole number of units such that the area of the rectangle is 24 square units. a. Which rectangle has the greatest perimeter? b. Which rectangle has the least perimeter?
Answer
We need pairs of whole numbers whose product is 24 — that is, the factor pairs of 24.
Rectangle (length × width)
Area
Perimeter = 2 × (l + w)
1 × 24
24 sq units
2 × 25 = 50 units
2 × 12
24 sq units
2 × 14 = 28 units
3 × 8
24 sq units
2 × 11 = 22 units
4 × 6
24 sq units
2 × 10 = 20 units
a. Greatest perimeter: the 1 × 24 rectangle, 50 units.
b. Least perimeter: the 4 × 6 rectangle, 20 units.
Why: the long thin 1 × 24 strip has two very long sides, so its boundary is huge. The 4 × 6 rectangle is the one closest to a square, and a square-like shape always has the shortest boundary for a given area.
Q6.
If you take a rectangle of area 32 sq cm, what will your answers be? Given any area, is it possible to predict the shape of the rectangle with the greatest perimeter as well as the least perimeter? Give examples and reasons for your answer.
Answer
Area 32 sq cm — the whole-number rectangles are:
Rectangle
Perimeter
1 cm × 32 cm
2 × 33 = 66 cm
2 cm × 16 cm
2 × 18 = 36 cm
4 cm × 8 cm
2 × 12 = 24 cm
Greatest perimeter: 1 cm × 32 cm (66 cm). Least perimeter: 4 cm × 8 cm (24 cm).
Yes — the shape can always be predicted:
The greatest perimeter comes from the longest, thinnest rectangle, that is 1 × (the area).
The least perimeter comes from the rectangle whose sides are closest to each other — the most square-like one. If the area is a perfect square, that rectangle is a square (area 36 → 6 × 6, perimeter 24).
Reason: for a fixed area, making one side longer forces the other to become shorter, but the long side grows much faster than the short side shrinks. So the perimeter shoots up as the rectangle gets thinner, and is smallest when the two sides are balanced.
Math Talk: this is why a farmer fencing a fixed area of land prefers a square field — it needs the least fencing and therefore costs the least.