What is the irregular shape area?

How it works

Choose how you describe the shape.

Corner coordinates (the shoelace formula). Type the corners (x₁, y₁), (x₂, y₂), …, (xₙ, yₙ) in order around the shape, 3 to 30 of them; after the last corner the edge returns to the first. The area is

A = ½ |Σ (xᵢ × yᵢ₊₁ − xᵢ₊₁ × yᵢ)|, for i = 1 to n, with (xₙ₊₁, yₙ₊₁) = (x₁, y₁).

Why it works: a triangle with corners P, Q and R has half the area of the parallelogram on PQ and PR, which is the size of their cross product (OpenStax Calculus Volume 3, Theorem 2.8). The cross product of (xᵢ, yᵢ) and (xᵢ₊₁, yᵢ₊₁) is xᵢyᵢ₊₁ − xᵢ₊₁yᵢ, positive when the turn from one to the next is counterclockwise and negative when clockwise, so the half-terms are the signed areas of the triangles from the origin to each edge. Going once around the shape, the parts outside it cancel and the sum is the area, positive or negative by the direction you went.

The perimeter is the sum of the edge lengths √((xᵢ₊₁ − xᵢ)² + (yᵢ₊₁ − yᵢ)²).

Triangles (Heron's formula). Split the shape into triangles that do not overlap, measure their three sides, and type each triangle as a row (1 to 20 triangles). Each triangle has area √(s(s − a)(s − b)(s − c)), with s = (a + b + c) ÷ 2, and the shape's area is their sum. The page computes it as √((a + b + c)(−a + b + c)(a − b + c)(a + b − c) ÷ 16), the same value.

Unit. Choose the unit the coordinates or sides are in. The area is in that unit squared; with any unit but "units (none)" the page also gives the area in m² (area × size²), ft² (m² ÷ 0.09290304) and hectares (m² ÷ 10,000). Sizes in metres: cm 0.01, m 1, km 1,000, in 0.0254, ft 0.3048, yd 0.9144, mi 1,609.344.

Rules

• Corners: a row's two cells are its x and y. A cell typed as a fraction (1/3) is read as that exact fraction, and a decimal as its exact value. No answer when two corners in a row are the same point, including the last and the first ("Two corners in a row are the same point. Each corner must differ from the next one."), or when the shape is not simple: two edges that do not share a corner touch or cross anywhere, or two edges in a row fold back over each other along one line ("Two edges cross or overlap. List the corners in order around the shape, so that no edge crosses another."). Corners in a straight line along one edge are allowed. The checks run on exact values.
• Triangles: each side is above 0 and at most 10⁹. No answer when a triangle's sides do not make a triangle, that is when a side is at least the sum of the other two ("Triangle k: each side must be shorter than the other two together, so a, b and c cannot make a triangle.", with k the row number and the sides as typed).
• A result too large or too small to show as a number gives no answer ("A result is too large or too small to show as a number.").

Output format

• The answer reads A u² (18.5 m²), or A square units with no unit, with A to 10 significant figures, rounded half up, thousands commas, and from 10¹⁵ up or below 10⁻⁶ as a × 10ⁿ. Every number output shows to 10 significant figures the same way.
• Typed numbers are read as exact decimals. The shoelace area, and the m², ft² and hectares from it, are exact fractions rounded once. Each edge length and each Heron area is exact when the fraction under the root is a square, and otherwise the true root rounded once to a 64-bit float; the perimeter and a total of triangle areas add these in order, in 64-bit floats once any term is a float.
• Steps are separated by semicolons. Corners: one step per edge, i to j: xᵢ × yⱼ − xⱼ × yᵢ = t, then Sum = S and Area = |S| ÷ 2 = A. Triangles: one step per triangle, Triangle k: s = (a + b + c) ÷ 2 = s, area = √(s × s−a × s−b × s−c) = A with each part written as its value, then Total area = A. Numbers in steps are exact values rounded once to 10 significant figures (or a float to 10), with the true minus sign (−) and no thousands commas.
• The answer sentence reads The area is A., with the answer text as above.

Worked examples by hand

A five-sided shape. Corners (0, 0), (4, 0), (5, 3), (2, 5), (0, 3), in metres:

• 1 to 2: 0 × 0 − 4 × 0 = 0
• 2 to 3: 4 × 3 − 5 × 0 = 12
• 3 to 4: 5 × 5 − 2 × 3 = 19
• 4 to 5: 2 × 3 − 0 × 5 = 6
• 5 to 1: 0 × 0 − 0 × 3 = 0

Sum = 37, so A = 37 ÷ 2 = 18.5 m². Perimeter 4 + √10 + √13 + √8 + 3 = 16.59625606 m.

A right triangle, listed clockwise. (0, 0), (0, 3), (4, 0) in feet: the terms are 0, −12 and 0, so the sum is −12 and A = |−12| ÷ 2 = 6 ft², which is 6 × 0.09290304 = 0.55741824 m². Perimeter 3 + 5 + 4 = 12 ft.

One triangle (OpenStax Example 5). Sides 10, 15 and 7: s = 16, A = √(16 × 6 × 1 × 9) = √864 = 29.39387691.

A lot (OpenStax Example 6). Sides 62.4, 43.5 and 34.1 m: s = 70, A = √(70 × 7.6 × 26.5 × 35.9) = √506,118.2 = 711.4198479 m², about 711.4 m² as in the example, or 0.07114198479 ha.

A four-sided lot and a diagonal. The 50 m diagonal splits it into triangles of 30, 40, 50 m and 35, 25, 50 m: √(60 × 30 × 20 × 10) = 600 m² and √(55 × 20 × 30 × 5) = √165,000 = 406.2019202 m², so A = 1,006.20192 m².