Trade Math II: Geometry, Trigonometry, and Area/Volume

Why Geometry and Trigonometry Matter in Surveying

Every measurement you record in the field becomes a number in a calculation. If the math is wrong, the map is wrong, the property line is wrong, and the earthwork bid is wrong. Trade Math II builds the geometric and trigonometric toolkit you will use daily as a survey technician, GIS analyst, or drone operator interpreting photogrammetric output. Accuracy is not optional: a 0.01-foot error in closure can fail a legal description, and a volume error in a cut/fill estimate can cost a contractor thousands of dollars.

Plane Geometry Fundamentals

A traverse is a series of connected lines (called courses) whose endpoints are survey stations. The interior angles of any closed polygon must satisfy a simple rule: the sum of interior angles equals (n − 2) × 180°, where n is the number of sides. For a four-sided traverse, the sum must be exactly 360°. When your measured angles do not sum to that value, the difference is the angular misclosure, which must be distributed (adjusted) among the measured angles before you can compute coordinates.

Bearings and azimuths both describe the direction of a course. A bearing is expressed as an angle from north or south toward east or west (e.g., N 42° 15′ E). An azimuth is a clockwise angle measured from north, ranging from 0° to 360°. Converting between them is a routine task: a bearing of S 42° 15′ W equals an azimuth of 180° + 42° 15′ = 222° 15′.

The Coordinate Method for Traverse Closure

Once angles are adjusted, you compute latitudes (north–south components) and departures (east–west components) for each course:

• Latitude = distance × cos(bearing angle)
• Departure = distance × sin(bearing angle)

For a perfectly closed traverse the sum of all latitudes equals zero and the sum of all departures equals zero. Any remaining sum is the linear misclosure, and the ratio of misclosure to total perimeter is the precision ratio (e.g., 1:5,000). The compass rule (Bowditch adjustment) distributes the misclosure proportionally to each course length. Once adjusted coordinates are known, the coordinate method (also called the double-meridian distance or shoelace formula) gives the enclosed area precisely.

Right-Triangle Trigonometry: SOH-CAH-TOA

For any right triangle with an acute angle θ:

• sin θ = opposite / hypotenuse
• cos θ = adjacent / hypotenuse
• tan θ = opposite / adjacent

A common field application is slope correction: if you tape a distance along ground that slopes at angle α, the horizontal distance = slope distance × cos α. On a 5% grade, α = arctan(0.05) ≈ 2° 52′, so the correction is small but accumulates over long distances.

Law of Sines and Law of Cosines

When triangles are not right triangles, two additional rules apply.

Law of Sines: a / sin A = b / sin B = c / sin C

Use this when you know two angles and one side (AAS or ASA), or two sides and the angle opposite one of them (SSA — use carefully; the ambiguous case can produce two solutions).

Law of Cosines: c² = a² + b² − 2ab cos C

Use this when you know three sides (SSS) or two sides and the included angle (SAS). A classic surveying use is computing an inaccessible baseline across a pond: you measure two accessible sides of a triangle and the included angle, then solve for the third side.

Average-End-Area Method for Earthwork Volumes

When a road or grading project is divided into cross-sections at regular intervals, the volume of material between two adjacent sections is:

V = L × (A₁ + A₂) / 2

where L is the distance between sections, A₁ and A₂ are the cross-sectional areas (in square feet or square meters), and V is volume (in cubic feet or cubic meters, then converted to cubic yards by dividing by 27). The average-end-area method slightly overestimates volume for converging sections; the prismatoid formula is more precise, but average-end-area is acceptable for many earthwork estimates and is the standard starting point for the CST exam.

Safety in the Field and at the Desk

Math errors carry real safety consequences. An incorrect volume calculation can lead to under-designed embankments that fail. When working outdoors on traverse fieldwork:

• Wear a high-visibility vest when near vehicle traffic; set up temporary traffic control per your agency's standards.
• Never set up on a slope without checking footing; a toppling total station injures people and destroys equipment.
• When using laptops or calculators near field equipment, keep liquids away and secure devices against wind.

At the computer, save field-book data and spreadsheet files regularly and maintain backups; data loss in professional practice is a serious liability issue.

Field-Book Discipline

All intermediate steps — angle sums, latitude/departure tables, adjustment corrections, area calculations — must appear in your field book in ink, with no erasures (cross out errors with a single line and initial). Entries should include the date, project name, instrument type, and your name. Professional surveyors are required by most state boards to maintain legible, complete records; the CST exam rewards this discipline.