Area — Free surveying reference · The Survey School

The hook

Three methods, one answer: coordinates (cleanest), DMD (the textbook hand method), and planimeter / Simpson's (for irregular shapes). The skill is picking the right tool for the polygon in front of you. Drag the vertices below to feel how area responds.

Area33350.0 sq units½ |Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|

Perimeter751.7 unitsΣ √((Δx)² + (Δy)²)

DirectionCCW (positive)sign of shoelace sum

Vertices (Easting, Northing)

#	E (x)	N (y)
1	60.0	60.0
2	280.0	70.0
3	290.0	220.0
4	50.0	200.0

Try this: drag the vertices into a long thin rectangle, then a near-square — same perimeter, very different area (a square always wins for max area at fixed perimeter). The shoelace sign flips when you drag a vertex to invert the polygon's winding — that's why the formula uses absolute value. On the exam you'll be asked to compute area from a coordinate table; run the shoelace by hand on a piece of scratch paper twice and you'll never forget it.

Memorize these

Concepts that show up on the exam

Coordinates method (shoelace)

2A = |Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)|. The default when you have a coordinate table — works for any closed polygon.

DMD (Double Meridian Distance)

A bookkeeping method using lat/dep. For each leg: DMD = previous DMD + previous dep + current dep. 2A = Σ (DMD · lat).

DPD (Double Parallel Distance)

Same idea as DMD but parallel to the X-axis instead of Y. Used when your reference line runs east-west.

Trapezoidal rule

For an irregular boundary along an offset line: A = h · (½y₁ + y₂ + … + y_{n−1} + ½y_n). Underestimates curves.

Simpson's 1/3 rule

Better for curved boundaries: A = (h/3) · (y₁ + 4·(y₂+y₄+…) + 2·(y₃+y₅+…) + y_n). Requires an even number of intervals.

Planimeter

A mechanical (or digital) tool that integrates area as you trace the boundary. Mostly historical now — you'll meet it in older texts and references, though almost nobody traces one in 2026.

Keep these in muscle memory

Formulas to know cold

Shoelace (coordinates)

A = ½ |Σᵢ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)|

Wraps around: index n+1 = index 1.

DMD

2A = Σ (DMD · lat)

DMD = double meridian distancelat = latitude of leg

Trapezoidal

A = h · (½y₁ + y₂ + … + ½yₙ)

Simpson's 1/3

A = (h/3) · (y₁ + 4Σodd + 2Σeven + yₙ)

Try it before you peek

Worked example

The problem

A four-corner parcel has corners at A(100,100), B(400,100), C(400,300), D(100,300). Find the area by the coordinates method.

Don't fall for these

What trips people up

Forgetting to close the polygon

The shoelace wraps from the last vertex back to the first. Skip that step and you compute the area of an open chain (which is meaningless).

Sign vs. area

Shoelace returns a SIGNED value: positive for counter-clockwise traversal, negative for clockwise. Always take the absolute value before reporting.

Mixing up lat/dep in DMD

DMD multiplies by LATITUDE (north component). Multiplying by departure gives DPD — different formula. The mnemonic: DMD goes with Meridian (north).

Test yourself

How well did it stick?

A quick 5-question check on Area. See where you stand and what to review.

Concepts that show up on the exam

Formulas to know cold

Worked example

What trips people up

How well did it stick?

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