Free · Plane geometry
Plane Shape Areas by Dimensions
Calculate areas for seventeen plane-shape families, retaining triangle, parallelogram, rhombus and kite measurement methods.
- Formula & worked example
- Private in your browser
- No signup
Calculator inputs
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How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- Use the worked example to check the method. Reset restores the starting fields.
Private by default
Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.
Formula and method
A square of side 3 m has area 9 m².
Use perpendicular height, side-angle trigonometry, diagonal cross products, circular sectors or regular-polygon triangle sums.
Worked example
Enter these known values and leave the other values blank.
- Shape
- Square
- Side length (a)
- 3 cm
- Square area
- 9 cm²
Assumptions and limitations
- Lengths are positive. Heights are perpendicular to the indicated base; ellipse inputs are semiaxes.
- A regular polygon has an integer number of sides of at least three. Pentagons, hexagons and octagons use equal sides and equal angles.
- Diagonal-and-angle area is half the product of diagonal lengths and the sine of their included angle, for a simple convex quadrilateral or parallelogram.
- Triangle ASA uses the side between the two positive angles, whose sum is below 180 degrees. SSS requires strict triangle inequalities.
- Inverse sine branches return the acute included angle; its supplementary angle can have the same area. ASA inverses return the angle in the open interval zero to 180 degrees.
- The retained soccer-field area unit means exactly 7140 m², a chosen 105 m by 68 m reference rectangle rather than every playing field.
Common questions
Why do diagonal formulas include a half?
Split a convex quadrilateral along a diagonal into two triangles. Their perpendicular heights add to the other diagonal times the sine of the angle between the diagonals.
References
Replace explanatory external images with original accessible descriptions preserving each scenario, rope or machine arrangement and every variant condition. Repair the missing one-half factor in both diagonal cross-product area formulas and their inverses, and replace constant-PI ASA angle inverses with the angle obtained from the cotangent identity. Restrict included angles to positive nondegenerate geometric ranges, circular sweeps to one full turn, and rhombus height to at most its side length. The international acre is exactly 43560 square international feet; Bookify replaces the rounded captured area factor.