Using Trigonometry to Find the Height of a Building

To estimate the height, H, of a building, measure the distance, D, from the point of observation to the base of the building and the angle, θ (theta), shown in the diagram. The ratio of the height H to the distance D is equal to the trigonometric function tangent θ (H/D = tan θ). To calculate H, multiply tangent θ by the distance D (H = D tan θ). The angle can be roughly estimated by pointing one arm at the base of the building and the other arm at the roof and judging whether the angle formed is close to 15°, 30°, 45°, 60°, or 75°. The angle can be estimated more accurately with a protractor and a plumb bob made of a pencil hanging from a string. Hang the plumb bob from the zero point in the middle of the straight edge of the protractor. Sight along the edge of the protractor at the roof of the building. Measure the angle formed by the straight edge of the protractor and the plumb bob. Subtract this angle from 90°.

Trigonometry, branch of mathematics that deals with the relationships between the sides and angles of triangles and with the properties and applications of the trigonometric functions of angles. The two branches of trigonometry are plane trigonometry, which deals with figures lying wholly in a single plane, and spherical trigonometry, which deals with triangles that are sections of the surface of a sphere.

The earliest applications of trigonometry were in the fields of navigation, surveying, and astronomy, in which the main problem generally was to determine an inaccessible distance, such as the distance between the earth and the moon, or of a distance that could not be measured directly, such as the distance across a large lake. Other applications of trigonometry are found in physics, chemistry, and almost all branches of engineering, particularly in the study of periodic phenomena, such as vibration studies of sound, a bridge, or a building, or the flow of alternating current.