Area Computations: Coordinates, Trapezoidal and Simpson's Rules

NCEES FE Civil spec: Surveying, B. Area computations. FE Reference Handbook: Civil Engineering chapter, Surveying section (area formulas).

What the FE Civil Spec Covers

Section B is about finding plan areas: parcels bounded by straight traverse lines, and strips bounded by an irregular line such as a stream or a ground profile. On the exam this means the coordinate (shoelace) method, the closely related double meridian distance (DMD) method, and the trapezoidal and Simpson's rules for equally spaced offsets. Problems are short but arithmetic heavy, so a clean table is your best tool.

Key Concepts

Closed traverse first. Area methods assume the traverse closes. If latitudes and departures do not sum to zero, balance them (for example with the compass rule) before computing area.
Coordinate method: cross multiply consecutive coordinates around the boundary. It works for any simple polygon, convex or not.
DMD method: uses latitudes and departures instead of coordinates; it gives the same double area as the coordinate method.
Offsets: perpendicular distances h measured at a common interval w from a straight base line to an irregular boundary.

Core Formulas

Area by coordinates for points 1 to n, taken in order around the parcel (point n+1 is point 1):

A = \frac{1}{2} i = 1 \sum n (x_{i} y_{i + 1} - x_{i + 1} y_{i})

An equivalent form multiplies each x by the difference of the neighboring y values:

A = \frac{1}{2} i = 1 \sum n x_{i} (y_{i + 1} - y_{i - 1})

Trapezoidal rule for n offsets at spacing w:

A = w (\frac{h _{1} + h _{n}}{2} + h_{2} + h_{3} + \dots + h_{n - 1})

Simpson's 1/3 rule (n odd):

A = \frac{w}{3} (h_{1} + 4 even k \sum h_{k} + 2 odd interior k \sum h_{k} + h_{n})

Where It Is in the FE Reference Handbook

Search the Civil Engineering chapter, Surveying section, for the area formulas: area by coordinates, the trapezoidal rule and Simpson's 1/3 rule all sit together there, just after the earthwork formulas. General polygon and triangle area formulas are in the Mathematics chapter.

Worked Example

A four-sided parcel has corners A (1000, 1000), B (1250, 1080), C (1300, 1320) and D (1060, 1260), given as (E, N) in meters. Find its area in square meters and hectares.

Step 1, shift the origin. Subtracting 1000 from every coordinate does not change the area and keeps the numbers small: A (0, 0), B (250, 80), C (300, 320), D (60, 260).

Step 2, cross products $x_{i} y_{i + 1} - x_{i + 1} y_{i}$ :

Pair	Computation	Result (m²)
A to B	0(80) minus 250(0)	0
B to C	250(320) minus 300(80)	56,000
C to D	300(260) minus 60(320)	58,800
D to A	60(0) minus 0(260)	0

Step 3, area.

A = \frac{1}{2} (0 + 56, 000 + 58, 800 + 0) = 57, 400 m^{2} = 5.74 ha

The sum is positive because the points run counterclockwise. Working with the original coordinates gives the same 114,800 m² double area, just with much larger intermediate numbers.

Exam Strategy

Before computing, sketch the parcel from the coordinates to confirm the points are in boundary order and to get a rough area estimate (for example, a bounding rectangle minus the corner triangles). If your result is far from the estimate, you have probably skipped a point or swapped a coordinate.

Shifting the origin, as in the example, is the single best time saver: subtract a round number from every easting and every northing so the cross products stay small. The area is unchanged because the shape is only translated.

For offset problems, count the offsets, not the intervals. Five offsets mean four intervals, which suits Simpson's 1/3 rule; six offsets mean five intervals, which does not. When a boundary is curved, Simpson's rule is usually closer to the true area, while the trapezoidal rule treats each strip as straight sided.

Common Mistakes

Listing the corners out of order, which produces a crossed polygon and a wrong area.
Forgetting to divide by 2 (the sum is the double area).
Mixing up N and E. Pick x = E and y = N and stay consistent; swapping only flips the sign.
Using Simpson's rule with an even number of offsets.
Unit slips: 1 acre = 43,560 ft², 1 ha = 10,000 m².

Practice Problems

Try these from the free PolyCalc FE Civil problem bank. Each one has a full worked solution.

What is the area of a rectangular parcel measuring 30 m by 20 m? See the worked solution or practice the Surveying deck
A triangular parcel has corners at (0, 0), (40, 0) and (10, 30), in metres. Its area is: See the worked solution or practice the Surveying deck
A closed traverse with a perimeter of 800 m has a closure error of +0.08 m in northing and −0.06 m in easting. The precision of the traverse is most nearly: See the worked solution or practice the Surveying deck

Related Calculators and Examples

Frequently Asked Questions

Does the order of the points matter in the coordinate area method?

Only for the sign. Going around the parcel counterclockwise gives a positive sum and clockwise gives a negative sum. The area is the absolute value, but the points must be in order around the boundary, never skipping across it.

When can I use Simpson's 1/3 rule?

Simpson's 1/3 rule needs equally spaced offsets and an odd number of offsets (an even number of intervals). If the count is even, use the trapezoidal rule, or apply Simpson to all but the last interval and the trapezoidal rule to that one.

How many square feet are in an acre and square meters in a hectare?

One acre is 43,560 square feet and one hectare is 10,000 square meters. FE answers for parcels are often given in acres or hectares, so convert at the end.

Back to all Surveying topics