NCERT Solutions for Class 8th Maths Chapter 1 Perfect Squares and Triangular Numbers; Square Roots — In-text Questions

Book page 7 Updated on2026-09-19

Q1.

Find how many numbers lie between two consecutive perfect squares. Do you notice a pattern?

Answer

Between n² and (n+1)² there are exactly 2n numbers.

Squares	Numbers strictly between	How many	2n
1 and 4	2, 3	2	2 × 1
4 and 9	5, 6, 7, 8	4	2 × 2
9 and 16	10 … 15	6	2 × 3
16 and 25	17 … 24	8	2 × 4

Why it happens: The gap itself is (n + 1)² – n² = 2n + 1. That count includes one of the two end points, so the numbers strictly in between number 2n + 1 – 1 = 2n. Notice the answer is always even, so no two consecutive squares are ever "close together" once n grows.

Tip: The official answer writes this as q – p – 1 for consecutive squares p and q. That is the same thing: with p = n² and q = (n+1)², q – p – 1 = (2n + 1) – 1 = 2n.

Q2.

How many square numbers are there between 1 and 100? How many are between 101 and 200? Using the table of squares you filled earlier, enter the values below, tabulating the number of squares in each block of 100.

1 – 100	101 – 200	201 – 300	301 – 400	401 – 500

501 – 600	601 – 700	701 – 800	801 – 900	901 – 1000

What is the largest square less than 1000?

Answer

Blockwise count of the perfect squares up to 1000:

Block	Squares in it	How many
1 – 100	1, 4, 9, 16, 25, 36, 49, 64, 81, 100	10
101 – 200	121, 144, 169, 196	4
201 – 300	225, 256, 289	3
301 – 400	324, 361, 400	3
401 – 500	441, 484	2
501 – 600	529, 576	2
601 – 700	625, 676	2
701 – 800	729, 784	2
801 – 900	841, 900	2
901 – 1000	961	1

Total = 10 + 4 + 3 + 3 + 2 + 2 + 2 + 2 + 2 + 1 = 31 squares below 1000, and the largest is

31² = 961 (the next square, 32² = 1024, is past 1000)

Why the counts thin out: The gaps between consecutive squares are 2n + 1, which keeps growing. Early on the gaps are 3, 5, 7 — many squares fit in a block of 100. Around 900 the gap is already about 60, so only one or two squares can fall in each block.

Q3.

Do you remember triangular numbers?

Page 7 — the first five triangular numbers.

Can you see any relation between triangular numbers and square numbers? Extend the pattern shown and draw the next term.

Page 7 — each square split into two triangular numbers; the last box is left empty for you.

Answer

Yes — two consecutive triangular numbers always add to a square.

1 + 3 = 4 = 2²
3 + 6 = 9 = 3²
6 + 10 = 16 = 4²
10 + 15 = 25 = 5² ← the next term
15 + 21 = 36 = 6²

So the next picture is a 5 × 5 square of dots split by a staircase into a triangle of 10 dots and a triangle of 15 dots.

A 5 × 5 square of dots cut into the triangular numbers 10 and 15.

Why it happens: The n-th triangular number is T_n = 1 + 2 + … + n = n(n+1)/2. Then
T_n–1 + T_n = (n–1)n/2 + n(n+1)/2 = n[(n–1) + (n+1)]/2 = n × 2n/2 = n².
Geometrically, two staircase triangles of heights n – 1 and n slot together into an n × n square.

Q4.

The area of a square is 49 sq. cm. What is the length of its side?

Answer

The side is the number that gives 49 when multiplied by itself.

side × side = 49
7 × 7 = 49, so side = 7 cm

We call 7 the square root of 49, and write √49 = 7.

Why we take only 7: Algebraically both 7 and –7 square to 49, so 49 has two integer square roots. A side length is a measured length, so only the positive root makes sense here. That is why this chapter works with the positive square root throughout.

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