11² - 9² = 121 - 81
= 40
Hence, 40 natural numbers lie between the squares of 11 and 9.

Introduction:
In this problem, we are required to find the number of natural numbers lying between 9 square and 11 square.

Solution:

We know that:

- 9 square = 81
- 11 square = 121

Therefore, we need to find the number of natural numbers lying between 81 and 121.

Method 1: Counting

We can count the number of natural numbers between 81 and 121 manually.

- 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120

There are 40 natural numbers between 81 and 121.

Method 2: Using Formula

We can also use a formula to find the number of natural numbers between two given numbers.

Number of natural numbers between two given numbers = (larger number - smaller number) - 1

Using this formula, we get:

- Number of natural numbers between 81 and 121 = (121 - 81) - 1
- Number of natural numbers between 81 and 121 = 40

Conclusion:

Therefore, there are 40 natural numbers lying between 9 square and 11 square.