A uniform square wooden board is partitioned into 9 smaller identical squares. Provided that the surface area of the original board measured 729 cm², calculate the length of the side of a single small square piece.

The correct option is 9 cm.

Step-by-Step Explanation:

Step 1: Calculate the surface area of a single small square piece.

We are given that the surface area of the original uniform square wooden board is 729 cm². The board is partitioned into 9 smaller identical squares.

Since all 9 smaller squares are identical, the surface area of one small square piece is obtained by dividing the total surface area of the board by 9:

$\text{Area of one small square}=\frac{729\text{ cm}{}^2}{9}=81\text{ cm}{}^2$

Step 2: Find the side length of the small square piece.

The area of a square is related to its side length by the formula:

$\text{Area}={\left(\text{side length}\right)}^2$

Taking the square root of both sides gives the side length of the small square piece:

$\text{Side length}=\sqrt{81}=9\text{ cm}$

Alternative Method (Using original board dimensions):

1. Find the side length of the original square wooden board by taking the square root of its area:

$\text{Original side length}=\sqrt{729}=27\text{ cm}$

2. Partitioning a square into 9 identical smaller squares forms a 3 × 3 grid. Thus, the side of the original board is divided into 3 equal lengths:

$\text{Side length of small square}=\frac{27\text{ cm}}{3}=9\text{ cm}$

Therefore, the side length of a single small square piece is 9 cm.