To find the volume of a triangular pyramid, we need the following information:

1. The area of the triangular base.
2. The height of the pyramid from the apex perpendicular to the base.

From the image provided, we have the following dimensions for the triangular base:

• The base of the triangle is $13 \, \text{cm}$.
• The height of the triangular base (perpendicular from the base to the opposite vertex) is $9 \, \text{cm}$.

Step 1: Calculate the Area of the Triangular Base

The area $A$ of a triangle is given by: $A = \frac{1}{2} \times \text{base} \times \text{height}$ Substituting the given values: $A = \frac{1}{2} \times 13 \, \text{cm} \times 9 \, \text{cm} = \frac{117}{2} = 58.5 \, \text{cm}^2$

Step 2: Use the Pyramid Volume Formula

The volume $V$ of a pyramid is given by: $V = \frac{1}{3} \times \text{Base Area} \times \text{Height}$ From the diagram, we see the height of the pyramid from the apex to the base is $5 \, \text{cm}$.

Substituting these values: $V = \frac{1}{3} \times 58.5 \, \text{cm}^2 \times 5 \, \text{cm} = \frac{292.5}{3} = 97.5 \, \text{cm}^3$

Answer

The volume of the triangular pyramid is: $\boxed{97.5 \, \text{cm}^3}$

Would you like further details on any of the steps or have any questions?

Here are five related questions to deepen your understanding:

1. How would the volume change if the height of the pyramid doubled?
2. What is the impact on the volume if the area of the triangular base is increased by 50%?
3. How do you calculate the volume if the base is not a triangle but a square or rectangle?
4. What formula would you use for the volume of a cone with a circular base?
5. How does the shape of the base affect the overall volume of pyramidal structures?

Tip: For any pyramid, the volume is always one-third of the product of its base area and height, making it unique from other volume calculations.