Triangle Area Calculator

Two roads lead to a triangle’s area: the base times the height, halved — or Heron’s formula, which needs only the three sides. Enter either the base and height, or all three sides, and this calculator returns the area (and, on the Heron road, the perimeter and a validity check).

Half a Parallelogram, and Heron’s Masterpiece

The base-height formula is a dissection: any triangle is half of the parallelogram on the same base and height, so A = b × h ÷ 2 — with the one condition students engrave wrongly, that h is the perpendicular height, the altitude, wherever it falls (outside the triangle entirely for obtuse ones, the formula unbothered). Heron’s formula is the deeper magic: from the three sides alone, with s the semiperimeter, A = √(s(s − a)(s − b)(s − c)) — no height measured, no angle known, the area conjured from the perimeter’s arithmetic. It is the surveyor’s formula (a field’s three taped sides yield its acreage in the office) and the calculator offers both roads with the agreement check built into its note, because a tool that computes one road silently teaches half the topic. The triangle-inequality guard is part of the lesson too: sides that cannot meet cannot enclose, and the page says so in the theorem’s own words.

Worked Examples: 30 by One Road, 84 by the Other

The defaults run both roads at once. Base-height: 10 × 6 ÷ 2 = 30 square units, the elementary figure. Heron: sides 13, 14, 15 give s = 21, and √(21 × 8 × 7 × 6) = √7056 = 84 exactly — the most famous Heronian triangle after 3-4-5’s family, its cleanness the signature of an integer-area integer-sided figure (the height to the side of 14 is exactly 12, so the first road confirms: 14 × 12 ÷ 2 = 84, the agreement the note celebrates). Examples engineered to agree are pedagogical gold: a reader can audit the machinery with mental arithmetic, then trust it on the unlovely numbers (a 7.3–9.1–11.6 field, say) where no audit but the formula’s correctness will serve. That trust, transferrable and earned, is the calculator’s product.

A Note on Measurement Error

Sides measured in the field carry error, and Heron’s formula transmits it unevenly: long thin triangles amplify mistakes (their area is the small difference of large quantities in the radicand), while near-equilateral figures forgive. A surveyor who re-tapes the long diagonal a second time rather than the short sides is practising this page’s numerics without stating them; a reader pricing land from a seller’s three distances should treat a sliver-shaped parcel’s quoted area as the estimate it is, and recompute from any better sides available.

Land, Sails and the Surveyor’s Chain

Heron’s road built the cadastral world: triangulating a plot into measured triangles and summing their areas is how land was taxed, sold and inherited for centuries — the triangle is the only polygon whose area its sides determine, rigid where quadrilaterals flex — and the method sails on in every mesh of computer graphics, where surfaces are triangulated and Heron (or its cross-product twin) prices each facet. Sailmakers cut by triangles and compute cloth by them; roofers meet them as gable ends, priced by base and height off a ladder. The calculator’s two roads are thus two professions’ habits preserved in one widget, and its inequality guard quietly teaches the rigidity fact the whole structure leans on: three sides fix a triangle utterly, area included — a property so useful that civilisation surveyed itself with it.

The Other Formulas in the Family

Two further roads deserve their map references: with two sides and the included angle, A = ½ab·sin C (the trigonometric form — the height in disguise, since the altitude to one side is the other side times the angle’s sine), and with coordinates, the shoelace of the FAQs. Equilateral triangles collapse everything to A = (√3 ÷ 4)a² — enter 6, 6, 6 in the Heron fields and read 15.59, the check that the family is one theory. A reader meeting an area problem in the wild identifies which data the problem supplies (a height, three sides, two sides and their angle, three coordinates) and takes the matching road; the calculator paves the two commonest and signposts the rest, which is the honest geography of the topic.

Related Tools

More from the Math Calculators hub: the Right Triangle Calculator — where the height is a leg and everything else opens too — and the Circumference Calculator, for the curved figure triangles approximate.

Frequently Asked Questions

Why must the height be perpendicular?

Because the dissection demands it: the parallelogram whose half the triangle is has area base × perpendicular height (its slanted side would overstate the count of unit squares), and the altitude is the measurement a plumb line makes from the opposite vertex to the base’s line. Using a slanted side as h is the classic error — it computes the area of a different, larger triangle — and obtuse triangles stress-test the understanding: their altitude to a short side falls outside the figure, on the base extended, and the formula still holds, which is why the definition says “the base’s line” and not “the base.” The calculator’s label spells “perpendicular” for exactly this reason.

What is a Heronian triangle?

One with integer sides and integer area — 3-4-5 (area 6), 13-14-15 (area 84), 5-5-6 (area 12) — a small aristocracy inside the integer-sided crowd, studied since antiquity because whole-number answers feel like messages. They are generated by rational constructions (pairs of Pythagorean triples sharing a leg, glued along it — the 13-14-15 is a 5-12-13 and a 9-12-15 sharing the altitude 12, which is exactly the agreement the worked example exhibits). The calculator recognises them by their cleanness; spotting one in the wild remains a small, reliable pleasure of the subject.

How do I get the area from coordinates?

By the shoelace formula — half the absolute sum of cross-products of consecutive vertices — or by splitting the figure into Heron triangles along a diagonal whose length the distance formula supplies. Both roads run through this hub’s instruments (the distance work on the right-triangle page, the areas here), and the shoelace is the surveyor’s actual office method for plotted land. The calculator’s side-based entry is the field half of that pipeline; enter the three taped or computed sides and the acreage follows, unit squares and all — divide by 43,560 for acres from square feet, the conversion every parcel eventually meets.

What if my triangle’s sides violate the inequality?

Then it is not a triangle: if a + b equals c exactly, the figure collapses to a segment (area zero, Heron’s radicand zero — the formula itself reports the degeneracy honestly), and if a + b is less than c, no arrangement of the three lengths closes, the radicand goes negative and the calculator’s guard speaks instead of a NaN. Field measurements fail this way when a side is misread or the plot is not planar (a hillside’s slope distances are longer than their map projections — surveyors reduce slope to horizontal before Heron ever sees the numbers). The guard is a data audit wearing a theorem’s clothes.

Is the area the same in any units?

The number scales with the square of the unit (a triangle of area 84 square feet is 12,096 square inches, the factor 144 being 12²) while the figure itself is indifferent — which is why the calculator reports in “square units of your input” and the conversions live in the hub’s conversion pages. The square-law scaling is also the deep reason similarity multiplies areas by the square of the length ratio — double a triangle’s sides and its area quadruples, as Heron’s formula shows on any example the reader cares to run twice through the fields above.

How do I find the height if I only know the sides?

From Heron’s area itself: once A is known, the altitude to any side is h = 2A ÷ that side — the two roads are one road seen from its ends (the worked example: 2 × 84 ÷ 14 = 12, the famous altitude). The maneuver matters because heights are what designs specify (a gable’s peak, a sail’s luff curve approximations) while sides are what fields supply; the calculator’s pair of outputs lets the reader cross between them at will. Direct altitude formulas exist (splitting the base by the adjacent sides’ difference of squares), but the area-mediated route is the one that never needs memorising — it is this page, run twice in spirit.

What is the largest area a given perimeter can enclose?

Among triangles, the equilateral’s — the isoperimetric theorem’s triangular case — which is why the equilateral earns its formula in the family section: for perimeter 18, sides 6-6-6 give 15.59, while a stretched 8-8-2 triangle manages only about 7.94 by Heron, and an isosceles 5-6.5-6.5 reaches 15.00 — close, and still short. Fencing a triangular garden on a fixed budget of edging is the domestic form of the theorem; run candidate side-triples through the Heron fields and the equilateral’s supremacy stops being a claim and becomes a measurement.

Who was Heron?

Heron of Alexandria (fl. 1st century AD), engineer of steam toys, automatic doors and surveying instruments, whose Metrica preserved the formula — probably older, plausibly Archimedes’ — with the first surviving proof-sketch, worked on a 7-8-9 triangle whose area (√(12 × 5 × 4 × 3) = √720 ≈ 26.83) he computes in the text. The calculator’s 13-14-15 example is chosen from his tradition of integer exemplars; his own example’s irrational answer makes the same point the guard makes — the formula prices every triangle that exists, and fame has not made it choosier.