To find the area of the triangular pyramid ABCD, we need to follow these steps:

Step 1: Find the length of leg BC.

We know that triangle ABC is a right triangle with:

• Vertical leg AB = 15
• Hypotenuse AC = 22

Using the Pythagorean theorem, we can find the length of leg BC:

$AC^2 = AB^2 + BC^2$ $22^2 = 15^2 + BC^2$ $484 = 225 + BC^2$ $BC^2 = 484 - 225 = 259$ $BC = \sqrt{259} \approx 16.1$

Step 2: Find the area of the triangular base.

The area $A_{\text{base}}$ of triangle ABC is given by:

$A_{\text{base}} = \frac{1}{2} \times AB \times BC$ $A_{\text{base}} = \frac{1}{2} \times 15 \times 16.1 = 120.75$

Step 3: Find the volume of the pyramid.

The volume $V$ of a pyramid is given by:

$V = \frac{1}{3} \times \text{Base Area} \times \text{Height}$

We already found the base area $A_{\text{base}} = 120.75$ and the height $BD = 28$. Now, we can calculate the volume:

$V = \frac{1}{3} \times 120.75 \times 28 = \frac{1}{3} \times 3381 = 1127$

Thus, the volume of the triangular pyramid is 1127 cubic units.

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Would you like more details or have any questions? Here are some related questions:

1. How do you calculate the volume of a pyramid with a square base?
2. What is the Pythagorean theorem, and how is it used in geometry?
3. How can you find the surface area of a triangular pyramid?
4. What is the difference between a triangular pyramid and a rectangular pyramid?
5. How can the area of any right triangle be calculated?

Tip: When using the Pythagorean theorem, always make sure that you're working with a right triangle (one angle is 90°), as the theorem only applies in that case.