If the area of a squared seminar hall is 729 m², find its length.

Step 1: Identify the given information.
The area of the squared seminar hall ($A$) is $729 \text{ m}^2$.

Step 2: Recall the formula for the side length of a square.
For a square, the area ($A$) is equal to the side length ($s$) squared ($A = s^2$). To find the side length, we take the square root of the area:

$$s = \sqrt{A}$$

Step 3: Substitute the given area into the formula.

$$s = \sqrt{729 \text{ m}^2}$$

Step 4: Calculate the square root of 729 using the long division method.
First, group the digits of 729 in pairs from the right: $\overline{7} \overline{29}$.

• Find the largest number whose square is less than or equal to the first group (7). $2^2 = 4$, which is less than 7. $3^2 = 9$, which is greater than 7. So, the first digit of the square root is 2. $$\begin{array}{r} 2 \\[-3pt] 2 \overline{| 7 \ 29} \\[-3pt] -4 \phantom{29} \\[-3pt] \hline 3 \phantom{29} \end{array}$$
• Bring down the next pair of digits (29) to the remainder (3), making it 329. Double the current quotient (2), which is $2 \times 2 = 4$. Write 4 as the first digit of the new divisor.
• Find the next digit for the divisor and quotient. We need a digit $x$ such that $4x \times x$ is less than or equal to 329. Let's try $x=7$: $47 \times 7 = 329$. $$\begin{array}{r} 2 \ 7 \\[-3pt] 2 \overline{| 7 \ 29} \\[-3pt] -4 \phantom{29} \\[-3pt] \hline 47 \overline{| 3 \ 29} \\[-3pt] -3 \ 29 \\[-3pt] \hline 0 \end{array}$$
• The remainder is 0, so the square root of 729 is 27.

Step 5: State the length of the seminar hall.

$$s = 27 \text{ m}$$

The length of the seminar hall is:

$$\text{27 m}$$