SamCalculator

Triangle Area Calculator

Find the area of a triangle using side lengths, base and height, Heron's formula, or vertex coordinates.

GeneralRightAreaIso · Equil

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Triangle area calculator

A triangle's area is half the product of any base and the perpendicular altitude to that base. From that one identity flow every other area formula — Heron's formula (from three sides), the SAS formula ½·ab·sin C (from two sides and the included angle), and the shoelace formula (from three vertex coordinates).

Pick whichever method matches the measurements you have. This tab handles all four standard cases and shows the working step-by-step so you can verify answers against textbook problems or hand-derive the formula yourself.

How each area method works

Base × height

Area = ½ · base · height. The simplest formula when you can measure or compute the perpendicular height directly. Works for any base — just make sure the altitude is perpendicular to that base.

Heron's formula

Given all three sides a, b, c, compute the semiperimeter s = (a+b+c)/2 and apply Area = √[s(s−a)(s−b)(s−c)]. No angles needed.

Two sides + included angle

Area = ½ · a · b · sin(C). The included angle is the one between the two given sides. Useful when you can't measure a perpendicular height.

Vertex coordinates (shoelace)

Area = ½ · |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|. Works for any three points in the plane — great for surveys, maps, and GIS data.

When to pick each method

1

Roof, slab, or floor plan

Use base × height when blueprints already give an altitude.

2

Lot bounded by GPS coordinates

Use the shoelace formula on the corner coordinates — no angles or distances required.

3

Triangle defined by three sticks

Use Heron's formula — measure each side and plug in.

4

Two sides framing a known angle

Use ½ ab sin C when the only available angle is the one between the two known sides.

5

Verifying textbook problems

Try the same triangle in two methods — the answers should match to within rounding.

6

Sanity check on irregular polygons

Decompose into triangles, area each one, and sum — a standard technique for computing polygon area.

Best practices

Always match base and height

If the base is side a, the height must drop to that side at 90° — not to side b or c.

Verify with the triangle inequality first

Heron's formula returns a complex number if a + b < c — the calculator catches this and shows an error.

Use radians inside trig calls

If you compute ½·ab·sin C by hand, convert C to radians before applying sin in most calculators.

Area formulas

Base × height

Area = ½ · base · height

Heron's formula

Area = √[s(s−a)(s−b)(s−c)]

Two sides + angle

Area = ½ · a · b · sin(C)

Shoelace (coordinates)

Area = ½ · |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|

Equilateral

Area = (√3 / 4) · a²

Inradius identity

Area = r · s

Common area mistakes

  • Forgetting the ½

    Both base×height and ab·sin C start with ½ — the area of the rectangle / parallelogram, halved.

  • Sign error in the shoelace formula

    If you list vertices clockwise, the unsigned area is positive but the formula returns negative. Always take the absolute value.

Frequently asked questions

If you know the base and the perpendicular height to that base, Area = ½ · base · height. Half the area of the rectangle (or parallelogram) you could draw around it. This is by far the most-used formula in practice when blueprints or measurements already supply an altitude.

Use it when you know all three side lengths but no angles or altitudes. Compute s = (a + b + c) / 2, then Area = √[s(s − a)(s − b)(s − c)]. It's exact and works for any triangle, but degenerate triangles (where two sides barely sum to the third) suffer from floating-point cancellation; in that case Kahan's variant Area = ¼·√[(a+b+c)·(-a+b+c)·(a-b+c)·(a+b-c)] is more accurate.

Use Area = ½ · a · b · sin C, where C is the angle between sides a and b. This is sometimes called the SAS area formula. It comes from dropping a perpendicular from one vertex onto the opposite side — that altitude is exactly b · sin C.

Given three vertices (x₁, y₁), (x₂, y₂), (x₃, y₃), the area is ½ · |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|. The absolute value is essential because the signed version returns negative area when the vertices are listed clockwise. The formula generalises to any polygon and is the standard way to compute the area of GIS or surveyed plots from their corner coordinates.

Yes — all four methods give the same area for the same triangle (to within floating-point rounding). Heron's, SAS, and shoelace are different algebraic paths through the same underlying geometry, while base × height is the definition the others reduce to. Trying the same triangle through two methods is a good way to catch input errors.

Internally, all arithmetic uses double-precision floating-point (≈15 significant digits). Displayed values are rounded to four decimal places by default — copy buttons keep six decimals. For most real-world surveys, blueprints, or homework problems this far exceeds the precision of the input measurements.

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