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NCERT Solutions for Class 9th Maths Chapter 6 In-text Questions — Perimeter of a Circle

Book page 120 Updated on2026-09-08

Q1.
Is the ratio of circumference (C) to diameter (D) the same for circles of all sizes (Fig. 6.5)? What do you think?
Answer

Yes, it is the same for every circle — that is exactly why the constant π exists.

Why it happens: any two circles are similar: enlarge the smaller one by the factor k = (bigger radius)/(smaller radius) and it becomes the bigger one exactly. An enlargement by factor k multiplies every length in the figure by k — the diameter and the circumference alike. So

C₂/D₂ = (kC₁)/(kD₁) = C₁/D₁.

The k cancels. The ratio therefore cannot depend on the size of the circle.

The same argument is what gives 4 : 1 for every square and 3 : 1 for every equilateral triangle. The difference is that for a square you can also see the answer by counting sides; for a circle you cannot, and the value of the ratio has to be hunted down.

Q2.
What is the value of the C/D ratio? How would you estimate this ratio?
Answer

The value is π = 3.14159265… — a number whose decimal expansion never ends and never repeats. There are two honest ways to estimate it: measure, or reason geometrically.

  • By measurement. Wrap a thread many times round a cylinder, unwind it, measure the total length, and divide by (number of turns × diameter). Wrapping 20 times instead of once divides your measuring error by 20 — this is why the book's home experiment says 20 turns.
  • By pure geometry. Trap the circle between two polygons. A regular hexagon inscribed in a circle of radius 1 has perimeter 6, and the circle is longer than it, so π = C/D > 6/2 = 3. Circumscribe a hexagon and you get an upper bound. Archimedes pushed this to 96-sided polygons and proved 3 10⁄71 < π < 3 1⁄7.
Did you know? Zu Chongzhi (480 CE) used a polygon with 24 576 sides to reach 355/113 ≈ 3.1415929, which stayed the world's best value for over 800 years.
Q3.
HOME MEASUREMENT: Take a cotton reel with thin thread around it. Measure the diameter D of the reel as accurately as possible. Unwrap and then tightly wrap the thread around the reel 20 times. Unwrap it again; measure its length L, and calculate L/(20D). This is the ratio we want. For accuracy, the thread should be very thin. Please do the experiment! Do you get a ratio between 3 and 4? Between 3.1 and 3.2?
Answer

You should get a value between 3 and 4, and with reasonable care between 3.1 and 3.2.

L = 20 × (circumference) = 20 × πD
∴ L / (20D) = π
Why it happens: twenty turns of thread lie one beside the other, each of length equal to one circumference. Dividing by 20 gives one circumference; dividing again by D gives the C/D ratio. The reason for using 20 turns rather than one is error control: if your ruler reading is off by 1 mm, that error is spread over twenty circumferences, so the error in π is twenty times smaller.
Tip: a thick thread wraps at a radius slightly larger than the reel's, and the extra thickness accumulates over 20 turns — which is why the book insists the thread be very thin. If your answer comes out noticeably above 3.2, suspect the thread, not the mathematics.
Q4.
It is also possible to estimate the C/D ratio using pure geometry, i.e., without any measurements at all! Can you imagine how?
Answer

Yes — by trapping the circle between polygons whose perimeters you can compute exactly.

Take a circle of radius 1, so D = 2 and C = 2π. Inscribe a regular hexagon. Joining the centre to two adjacent vertices makes a triangle with two sides equal to 1 and the angle between them 360°/6 = 60°; so the triangle is equilateral and each side of the hexagon is 1.

perimeter of inscribed hexagon = 6 × 1 = 6
the circle is longer, so 2π > 6
∴ π > 3

Now circumscribe a regular hexagon (each side touching the circle). Half of one side, the radius to the point of contact and the line to a vertex form a right triangle with a 30° angle, and the Baudhāyana–Pythagoras theorem gives half-side = 1/√3. So each side is 2/√3 and

perimeter of circumscribed hexagon = 6 × 2/√3 = 12/√3 = 4√3
the circle is shorter, so 2π < 4√3
∴ π < 2√3 ≈ 3.46
Why it happens: a convex curve that lies inside another convex closed curve is shorter than it. The inscribed hexagon lies inside the circle and the circle lies inside the circumscribed hexagon, so the three perimeters are in increasing order. Doubling the number of sides tightens both bounds, and that is precisely Archimedes' method — with 96 sides he squeezed π between 3.1408 and 3.1429.
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