Area of 2D Shapes Calculator: Heron, Trapezoids and Sectors
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Area of 2D Shapes Calculator
Area and perimeter for the shapes that actually turn up in takeoff work, including the trapezoid, the circle sector and regular polygons. Triangles can be solved from base and height or from three sides.
Units are whatever you enter: metres in gives square metres out. The three sided triangle uses Heron. The regular polygon takes the number of sides and the side length, and the sector angle is in degrees.
How to use this tool
Pick the shape, and the input boxes relabel themselves to what that shape needs. The result is the area in whatever square unit matches your inputs, with the perimeter or circumference alongside where it has a clean answer.
Two triangles are on the list deliberately. If you know the base and the perpendicular height, use that mode. If all you have is three side lengths, which is the usual case when you have taped an irregular parcel, use the three sided mode and the tool applies Heron. If the three sides cannot form a triangle at all, it says so instead of returning a confident nonsense number.
The formulas, and where each one earns its keep
The rectangle, parallelogram and base height triangle are all the same idea: base times perpendicular height, halved for the triangle. The word perpendicular is doing the work there. The slant side of a parallelogram is not its height, and using it overstates the area by exactly the factor people then cannot find.
The trapezoid is the average of the two parallel sides times the height between them, which is why it turns up constantly in earthworks: a cutting cross section with a flat base and battered sides is a trapezoid, and end area volume methods are just this formula applied twice.
The regular polygon formula, n s squared over 4 tan of pi over n, covers everything from an equilateral triangle to effectively a circle. At six sides and side length 1 it returns 2.598, and as the side count climbs the answer converges on pi r squared, which is a satisfying check that the two formulas agree about the world.
A worked check you can repeat on paper
A parcel taped as a triangle with sides 3, 4 and 5. The semi perimeter is 6, and Heron gives the root of 6 times 3 times 2 times 1, which is the root of 36, exactly 6. Since 3-4-5 is a right triangle, the base height route gives half of 3 times 4, also 6. Two different formulas, one answer, which is precisely how this tool was verified before publishing.
Mistakes that catch people out
Using the slant side as the height. Parallelograms and trapezoids need the perpendicular distance, not the sloping edge. Fix: if you taped along the slope, you have the hypotenuse, and the height comes from Pythagoras.
Averaging the wrong pair on a trapezoid. The average is of the two parallel sides only. Fix: identify which pair is parallel before averaging.
Mixing units. A base in metres and a height in millimetres gives an area in nothing at all. Fix: one unit throughout, converted before entry, not after.
Trusting a taped triangle that cannot close. Sides of 2, 3 and 8 do not meet, and a taping error is the usual culprit. Fix: the tool rejects these, so re-tape rather than force it.
How this was checked
Every formula was verified against exact cases before this page went live. Heron on 3-4-5 returns exactly 6 and agrees with the base height route. The hexagon of side 1 returns 3 root 3 over 2. The square run through the polygon formula at n equals 4 returns the side squared. A 90 degree sector returns exactly a quarter of its circle, and the ellipse with equal axes returns exactly the circle area.
Related Calculators and Reference
For volumes rather than areas, the Volume of 3D Shapes Calculator extends these sections into solids. The three sided triangle connects to the full Triangle Solver when you also need the angles. Land areas in acres, roods and perches convert in the Area Converter, and cross section based earthwork volumes live in the Earthworks Volume Calculator.