To solve the problem, we need to calculate the volume of a triangular pyramid (S.LPT) where:

• The base $LPT$ is an equilateral triangle with side length $66 \, \text{mm}$.
• The height of the pyramid $SO = 48 \, \text{mm}$.
• $O$ is the centroid of the base triangle $LPT$.
• The height of triangle $LPT$ is $OH = 11\sqrt{3} \, \text{mm}$.

Formula for the Volume of a Pyramid:

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

• $A_{\text{base}}$ is the area of the base (triangle $LPT$).
• $h = SO = 48 \, \text{mm}$ is the height of the pyramid from $S$ to the base $LPT$.

Step 1: Calculate the Area of Triangle $LPT$

The area of an equilateral triangle is: $A_{\text{base}} = \frac{s^2 \sqrt{3}}{4}$ Where $s = 66 \, \text{mm}$.

Step 2: Use the Pyramid Volume Formula

Substitute $A_{\text{base}}$ and $h$ into the volume formula to get the result.

I will now calculate the solution.The volume of the triangular pyramid $S.LPT$ is approximately $30179.3 \, \text{mm}^3$.

Do you need further clarification or details on any of the steps?

Here are some related questions you might consider:

1. How is the centroid of a triangle calculated?
2. What are other methods for calculating the volume of pyramids?
3. How does the area of an equilateral triangle relate to its side length?
4. What is the significance of $OH = 11\sqrt{3} \, \text{mm}$ in this problem?
5. How would the volume change if the height of the pyramid doubled?

Tip: When working with three-dimensional objects, always confirm if the base area and the height are perpendicular for accurate volume calculation.