To determine the height of a pyramid using trigonometry, you need to measure the base length and an angle, then apply trigonometric ratios.

Start by measuring the length of one side of the pyramid’s base, as well as the angle between the base and the slant height. The slant height is the line that extends from the apex of the pyramid to the midpoint of one side of the base. This angle is commonly referred to as the angle of elevation.

Next, you can utilize the tangent function, which connects the angle of elevation to the opposite side (the height of the pyramid) and the adjacent side (half the length of the base). The relationship is expressed by the formula:

To find the height, rearrange this formula:

For instance, if the length of the base of the pyramid is $10$ metres and the angle of elevation is $30$ degrees, you would first calculate half the base length, which is $5$ metres. Then, using the tangent function, you can write:

Since $\tan(30^\circ) \approx 0.577$, we can substitute this value into the equation:

Therefore, the height of the pyramid is approximately $2.885$ metres. This method enables you to calculate the height using fundamental trigonometric principles and straightforward measurements.