Trigonometry in three dimensions

The angle between a sloping line and a horizontal plane is the angle between that line and its horizontal projection. Identify the foot of the perpendicular, then draw the right-triangle section separately.

Example

A vertical pole \(AT\) of height 8 m stands at corner \(A\) of a horizontal rectangle \(ABCD\), with \(AB=12\text{ m}\) and \(BC=9\text{ m}\). The projection of \(CT\) is \(CA\), and \(CA=\sqrt{12^2+9^2}=15\text{ m}\). Therefore \(\theta=\angle TCA=\tan^{-1}(\dfrac8{15})\approx28.1^\circ\).

The line of sight has length \(CT=17\text{ m}\). Using 12 m or 9 m as the adjacent side would measure a different section. In a pyramid, a section through a base corner and a section through a side midpoint generally have different angles.