To solve the problem, we are asked to calculate the volume of a right pyramid with a square base.

Given:

• The base of the pyramid is a square with side length $9 \, \text{m}$.
• The height of the pyramid is $6 \, \text{m}$.

Formula for the volume of a pyramid:

The volume $V$ of a pyramid is given by the formula: $V = \frac{1}{3} \times \text{Area of the base} \times \text{Height}$

Step 1: Calculate the area of the base

The area of a square is given by: $\text{Area of the base} = \text{side}^2 = 9^2 = 81 \, \text{m}^2$

Step 2: Apply the formula for volume

Now, we substitute the area of the base and the height into the formula for the volume: $V = \frac{1}{3} \times 81 \, \text{m}^2 \times 6 \, \text{m}$

Step 3: Simplify the calculation

$V = \frac{1}{3} \times 486 \, \text{m}^3 = 162 \, \text{m}^3$

Answer:

The volume of the pyramid is $162 \, \text{m}^3$.

Thus, the correct choice is: D) 162 $\text{m}^3$.

Would you like more details, or do you have any questions?

Here are five related questions:

1. How do you calculate the volume of a pyramid with a triangular base?
2. What is the significance of the height in the formula for the volume of a pyramid?
3. How does the formula for the volume of a pyramid compare to that of a cylinder?
4. If the height of the pyramid were doubled, how would that affect the volume?
5. What are the units of the volume and why?

Tip: Remember that the volume of a pyramid is always one-third of the product of the base area and the height, which is why pyramids have less volume than prisms of the same base and height.