# What is the irregular shape area?

Finds the area of an irregular shape from its corner coordinates by the shoelace formula, or from triangles of three measured sides by Heron’s formula, with the perimeter, square metres, square feet and hectares.

- Page: https://www.acalculator.org/math/irregular-shape-area-calculator
- JSON spec: https://www.acalculator.org/math/irregular-shape-area-calculator.json
- Version: c68579f48ebb

## Default answer

Example with the default inputs (I know Corner coordinates, Corners (x, y), in order around the shape [0, 0; 4, 0; 5, 3; 2, 5; 0, 3], Length unit m): The area is 18.5 m².

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| by | I know | The corners as x and y coordinates, or the shape split into triangles with three measured sides each. |
| p | Corners (x, y), in order around the shape | One corner per row, x then y, going around the shape in order: 3 to 30 corners. |
| t | Triangles | Split the shape into triangles that do not overlap, and type the three sides of each. |
| u | Length unit | The unit of the coordinates or sides; with a unit, the area is also shown in m², ft² and hectares. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| result | Area | The area in the chosen unit squared. |
| area | Area (unit²) | The area in the typed unit squared. |
| perimeter | Perimeter (unit) | The distance around the shape: the sum of the edge lengths. |
| m2 | Area, m² | The area in square metres. |
| ft2 | Area, ft² | The area in square feet: m² ÷ 0.09290304. |
| hectares | Area, hectares | The area in hectares: m² ÷ 10,000. |
| steps | Steps | The area worked out, step by step. |

## Method

Corners: A = ½ |Σ (xᵢ × yᵢ₊₁ − xᵢ₊₁ × yᵢ)| (the shoelace formula), going around the corners and back to the first. Triangles: each area is √(s(s − a)(s − b)(s − c)) with s = (a + b + c) ÷ 2 (Heron’s formula), and the areas add up.

## Assumptions

- The corners are listed in order around the shape, and its edges do not cross.
- Triangles do not overlap and together cover the shape.
- Typed numbers are read as exact decimals; a root that is not exact is rounded once.

## Worked examples

1. by = corners, p = 0 or 0 or 4 or 0, u = m gives area = 18.5, result = 18.5 m², steps = 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; Area = |37| ÷ 2 = 18.5, perimeter = 16.59625606. Source: OpenStax, Calculus Volume 3, §2.4 The Cross Product (Theorem 2.8: ‖u × v‖ is the area of the parallelogram with sides u and v; Example 2.39: a triangle’s area is half of it), https://openstax.org/books/calculus-volume-3/pages/2-4-the-cross-product (retrieved 2026-10-19).
2. by = corners, p = 0 or 0 or 0 or 3, u = ft gives area = 6, perimeter = 12, ft2 = 6, m2 = 0.55741824. Source: OpenStax, Calculus Volume 3, §2.4 The Cross Product (Theorem 2.8: ‖u × v‖ is the area of the parallelogram with sides u and v; Example 2.39: a triangle’s area is half of it), https://openstax.org/books/calculus-volume-3/pages/2-4-the-cross-product (retrieved 2026-10-19); NIST Special Publication 811, Appendix B.8 (1 in = 0.0254 m, 1 ft = 0.3048 m, 1 ft² = 0.09290304 m², 1 yd = 0.9144 m, 1 mi = 1,609.344 m, 1 ha = 10,000 m²), https://www.nist.gov/pml/special-publication-811/nist-guide-si-appendix-b-conversion-factors/nist-guide-si-appendix-b8 (retrieved 2026-10-19).
3. by = triangles, t = {"a":10,"b":15,"c":7} or undefined, u = units gives area = 29.39387691, result = 29.39387691 square units. Source: OpenStax, Precalculus 2e, §8.2 Non-right Triangles: Law of Cosines (Heron’s formula, Area = √(s(s − a)(s − b)(s − c)) with s = (a + b + c) ÷ 2; Example 5: sides 10, 15 and 7 give about 29.4; Example 6: a lot with sides 62.4 m, 43.5 m and 34.1 m has about 711.4 m²), https://openstax.org/books/precalculus-2e/pages/8-2-non-right-triangles-law-of-cosines (retrieved 2026-10-19).
4. by = triangles, t = {"a":62.4,"b":43.5,"c":34.1} or undefined, u = m gives area = 711.4198479, hectares = 0.07114198479. Source: OpenStax, Precalculus 2e, §8.2 Non-right Triangles: Law of Cosines (Heron’s formula, Area = √(s(s − a)(s − b)(s − c)) with s = (a + b + c) ÷ 2; Example 5: sides 10, 15 and 7 give about 29.4; Example 6: a lot with sides 62.4 m, 43.5 m and 34.1 m has about 711.4 m²), https://openstax.org/books/precalculus-2e/pages/8-2-non-right-triangles-law-of-cosines (retrieved 2026-10-19).
5. by = triangles, t = {"a":30,"b":40,"c":50} or {"a":35,"b":25,"c":50}, u = m gives area = 1,006.20192. Source: OpenStax, Precalculus 2e, §8.2 Non-right Triangles: Law of Cosines (Heron’s formula, Area = √(s(s − a)(s − b)(s − c)) with s = (a + b + c) ÷ 2; Example 5: sides 10, 15 and 7 give about 29.4; Example 6: a lot with sides 62.4 m, 43.5 m and 34.1 m has about 711.4 m²), https://openstax.org/books/precalculus-2e/pages/8-2-non-right-triangles-law-of-cosines (retrieved 2026-10-19).

## FAQ

### How do you calculate the area of an irregular shape?

Split it into pieces whose area you know and add them. With corner coordinates, the shoelace formula does it in one step: A = ½|Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|. The shape with corners (0, 0), (4, 0), (5, 3), (2, 5), (0, 3) has A = ½ × 37 = 18.5.

### How does the shoelace formula work?

Go around the corners in order. For each edge, multiply the x of its first corner by the y of the next, and subtract the x of the next times the y of the first. Add the results, take the size and halve it. Each term is twice the signed area of a triangle with the origin, so the overlaps cancel.

### How do I find the area of an irregular lot from its side lengths?

Side lengths alone do not fix the shape, so measure a diagonal too. It splits a four-sided lot into two triangles, each with three known sides: Heron’s formula gives each area. Sides 30, 40 and 50 m give 600 m²; 35, 25 and 50 m give 406.2 m²; the lot is 1,006.2 m².

### What is Heron’s formula?

Area = √(s(s − a)(s − b)(s − c)), where s = (a + b + c) ÷ 2 is half the perimeter (OpenStax Precalculus 2e §8.2). A triangle with sides 10, 15 and 7 has s = 16 and area √(16 × 6 × 1 × 9) = √864 ≈ 29.39.

### Does the order of the corners matter?

Yes: list them in order around the edge, clockwise or counterclockwise. A jumbled order makes edges cross, and the shoelace formula then gives the wrong area, so the page asks you to fix the order.

### Can I get the area in square feet or square metres?

Yes. Choose the unit your measurements are in: the page gives the area in that unit squared and also in m², ft² (1 ft² = 0.09290304 m²) and hectares (1 ha = 10,000 m²).

## Sources

- OpenStax, Calculus Volume 3, §2.4 The Cross Product: Theorem 2.8 (‖u × v‖ is the area of the parallelogram with adjacent sides u and v) and Example 2.39 (the triangle PQR has half that area), retrieved 2026-10-19. https://openstax.org/books/calculus-volume-3/pages/2-4-the-cross-product
- OpenStax, Precalculus 2e, §8.2 Non-right Triangles: Law of Cosines: Heron’s formula, Area = √(s(s − a)(s − b)(s − c)) with s the semi-perimeter; Example 5 (sides 10, 15 and 7: about 29.4 square units) and Example 6 (a lot with sides 62.4 m, 43.5 m and 34.1 m: about 711.4 m²), retrieved 2026-10-19. https://openstax.org/books/precalculus-2e/pages/8-2-non-right-triangles-law-of-cosines
- NIST Special Publication 811, Appendix B.8: 1 in = 0.0254 m, 1 ft = 0.3048 m, 1 ft² = 0.09290304 m², 1 yd = 0.9144 m, 1 mi = 1,609.344 m, 1 ha = 10,000 m², all exact, retrieved 2026-10-19. https://www.nist.gov/pml/special-publication-811/nist-guide-si-appendix-b-conversion-factors/nist-guide-si-appendix-b8