Irregular Lawn Area Calculator

How it works

The offset method measures a curved area the way surveyors do. Run a straight baseline (a tape or string) through the longest part of the area. At equal intervals along it, measure the width of the area square to the baseline; these widths are the offsets. Where the edge meets the baseline at either end the offset is 0.

The trapezoidal rule treats each strip between two offsets as a trapezoid: the area is the interval times the sum of the offsets, with the two end offsets counted at half weight. When both ends are 0 this is simply the interval times the sum of the offsets. Simpson's rule fits a curve through each set of three offsets and is usually closer for smooth, rounded edges; it needs an odd number of offsets, so that the strips come in pairs.

The radius method suits roughly round areas. From a point inside the area, measure the distance to the edge in evenly spaced directions all the way round. Each radius stands for a thin wedge whose area is half the radius squared times its angle, so the area is π times the average of the squared radii (the polar form of the trapezoidal rule). Squaring before averaging matters: a circle measured from a point off its center gives some long and some short radii, and the average of their squares still gives the circle's exact area, where squaring the average radius would read low.

Formula

trapezoidal: A = h × (o₀/2 + o₁ + o₂ + … + oₙ₋₁ + oₙ/2)
Simpson's:   A = h/3 × (o₀ + 4o₁ + 2o₂ + 4o₃ + … + 4oₙ₋₁ + oₙ)   (n even)
radii:       A = ½ × Σ rᵢ² × (2π ÷ n) = π × average of rᵢ²   (n radii at equal angles)
h = interval between offsets; o₀ … oₙ = offsets
acres = sq ft ÷ 43,560

Example

Offsets of 0, 14, 22, 25, 23, 16 and 0 ft taken every 10 ft along a 60 ft baseline: the trapezoidal rule gives 10 × (0 + 14 + 22 + 25 + 23 + 16 + 0) = 1,000 sq ft (0.022957 acres, 1 unit of 1,000 sq ft). Simpson's rule gives 10/3 × (0 + 56 + 44 + 100 + 46 + 64 + 0) = 1,033.33 sq ft.

Eight radii of 20, 24, 22, 18, 21, 23, 19 and 25 ft have squares averaging (400 + 576 + 484 + 324 + 441 + 529 + 361 + 625) ÷ 8 = 467.5 sq ft, so the area is π × 467.5 = 1,468.69 sq ft.

Assumptions and limitations

• Offsets are measured square to a straight baseline at exactly equal intervals, and the edge between stations is smooth. Closer stations follow a wiggly edge better.
• Both rules are approximations. The trapezoidal rule cuts across curves with straight lines; Simpson's rule is exact for edges that are parabolic between stations and usually closer for rounded shapes.
• The radius method assumes the radii are taken at equal angles all the way round and that each ray from the measuring point crosses the edge only once, as it does for roughly round shapes. It gives a circle's exact area even from an off-center point; elongated or lobed shapes still read approximately, more closely with more radii.
• The area is measured flat (in plan); a sloping lawn has more surface than its plan area.

Frequently asked questions