NCERT Solutions for Class 9th Maths Chapter 6 In-text Questions — Area of a Circle
Book page 146 Updated on2026-09-08
Q1.
As the slices become smaller and smaller, the arcs in Fig. 6.37B become more and more closer to a line. This makes the figure more and more closer to a parallelogram with base = half the circumference (why?) = πr, height = radius r.
Answer
Because the slices are laid out alternately — one pointing up, the next pointing down — so exactly half of them contribute their arcs to the top edge and the other half to the bottom edge.
Cut the disc into 2n equal slices (sectors). Every slice has two straight sides of length r and one arc. All 2n arcs together = the whole circumference = 2πr.
Arrange them alternately point-up, point-down. The n point-up slices lay their arcs along the bottom edge. The n point-down slices lay their arcs along the top edge.
length of top edge = n arcs = ½ × 2πr = πr height = the straight side of a slice = r
Area of the disc = area of the near-parallelogram = base × height = πr × r = πr²
Alternate slices point up and down, so the arcs split evenly between the two long edges — each edge gets half the circumference.
Why it happens: nothing is created or lost by rearranging the slices, so the area is unchanged. What changes is the shape, and it approaches a parallelogram: as the slices get thinner, each arc gets closer to a straight segment and the slanting ends get closer to vertical. Nīlakaṇṭha Somayājī gave exactly this picture around 1500 CE, in his commentary on the Āryabhaṭīya. It is a limiting argument — the first idea of calculus — and it is why the answer is exactly πr² rather than an approximation.
Check it yourself: the same picture also explains Archimedes' statement that a circle equals a right triangle with legs r and 2πr — the parallelogram of base πr and height r has the same area as a triangle of base 2πr and height r.