To determine the height of a triangle in three-dimensional space, we can utilize trigonometric ratios that involve the angles and side lengths of the triangle.

Let’s delve into the details. A 3D triangle is defined as a triangle situated in three-dimensional space. To find its height, we first need to identify a base and the corresponding height from a vertex perpendicular to this base. For instance, consider a triangle with vertices labeled as $A$, $B$, and $C$. If we wish to find the height from vertex $A$ perpendicular to the base $BC$, we can follow these steps:

1. Calculate the Length of Base $BC$: If we have the coordinates of points $B$ and $C$, we can apply the distance formula to find the length of the base $BC$:

   $$BC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$

2. Determine the Angle at Vertex $A$: We will denote this angle as $\angle BAC$. If we know the lengths of sides $AB$ and $AC$, we can employ the cosine rule to compute this angle:

   $$\cos(\angle BAC) = \frac{AB^2 + AC^2 - BC^2}{2 \cdot AB \cdot AC}$$

3. Calculate the Height from $A$ to Base $BC$: With the angle $\angle BAC$ determined, we can use the sine function to find the height $h$ from vertex $A$ to the base $BC$:

   $$h = AB \cdot \sin(\angle BAC)$$

Alternatively, if the area of triangle $ABC$ and the length of base $BC$ are known, we can utilize the formula for the area of a triangle:

By rearranging this formula, we can solve for $h$:

These methods provide a systematic approach to finding the height of a triangle in 3D space using trigonometric principles, leveraging angles, side lengths, and the area of the triangle.