// maths › Navigation & Aviation
A worked real-world example: a ground controller measures the angle of elevation to an aircraft a known horizontal distance away, then uses tangent to find its altitude and cosine to find the slant (radar) range — shown on an animated sky scene you can adjust.
altitude = ground distance × tan θ
A mind behind this: Hipparchus of Nicaea c. 190 – c. 120 BC
If you know the horizontal ground distance to the point beneath the aircraft and measure the angle of elevation, the altitude is that distance times the tangent of the angle — the plane sits at the top of a right triangle.
Altitude is the vertical height of the aircraft. Slant range is the straight-line distance from the observer to the plane — the hypotenuse — which a radar actually measures. Cosine links the ground distance to the slant range.
The underlying radar and tracking systems do it automatically, but the geometry is exactly this: angles and distances are continuously turned into positions, heights and ranges. The maths is the same as this worked example.
Radar gives slant range and bearing; trigonometry converts those into the separate altitude and ground-distance figures controllers and pilots need. Each ratio (sine, cosine, tangent) extracts a different part of the same triangle.
Air-traffic control, military radar, missile and drone tracking, and even amateur rocketry all turn a measured angle and a known distance into height and range. Surveyors and ballooning teams use the identical triangle.