Triangle Area Calculator

This calculator finds the area of a triangle from its three side lengths using the Heron formula, A = √(s(s−a)(s−b)(s−c)), where s is half the perimeter. Enter three sides above and it shows every step of the substitution.

Two ways to find a triangle area

If you know the base and the perpendicular height, the area is ½ × base × height. That is the formula most people meet first, and it is the quickest when a height is available.

Real triangles rarely come with a height marked on them. A plot of land, a roof panel or a sail has three measurable sides and no obvious perpendicular. That is where the Heron formula earns its place: it needs nothing but the three lengths.

Why half base times height works

Any triangle is exactly half of the parallelogram formed by duplicating it and rotating the copy 180 degrees. The parallelogram has area base times height, so the triangle has half of it. The height must be perpendicular to the base, not along a slanted side, which is the same trap that catches people with cones.

A worked example with Heron

Take sides of 30, 45 and 50 units.

• Half the perimeter: s = (30 + 45 + 50) / 2 = 62.5
• Subtract each side: 62.5 − 30 = 32.5, 62.5 − 45 = 17.5, 62.5 − 50 = 12.5
• Multiply: 62.5 × 32.5 × 17.5 × 12.5 = 444,335.9375
• Take the square root: 666.5853 square units

Heron of Alexandria described this around AD 60, though there is evidence Archimedes knew it earlier. It remains one of the more surprising results in elementary geometry, because nothing about the formula hints at where the square root comes from.

When three sides do not make a triangle

Any two sides must add to more than the third. Sides of 3, 4 and 10 cannot close: the short pair laid end to end still falls short of the long one. Feed such a set into the Heron formula and the product goes negative, leaving you taking the square root of a negative number.

The calculator checks this before computing and tells you rather than returning a meaningless answer. It is worth knowing as a sanity check on measurements taken in the field, since a set of three lengths that cannot form a triangle usually means one was mistyped.

Right triangles are simpler

If one angle is 90 degrees, the two shorter sides are perpendicular to each other, so one is the base and the other is the height. The area is simply half their product. A triangle with legs of 3 and 4 has an area of 6, and by Pythagoras its third side is 5.

The 3-4-5 relationship is the oldest surveying tool still in use. Knot a rope at those intervals, pull it taut into a triangle, and the angle between the short sides is exactly 90 degrees. Builders still square out foundations this way, usually scaled up to 3, 4 and 5 metres.

Other routes to the area

What you know	Formula
Base and perpendicular height	½bh
Three sides	Heron
Two sides and the angle between	½ab sin C
Three corner coordinates	The shoelace formula

The two-sides-and-an-angle version is the surveyor's workhorse, because a theodolite measures angles directly. The shoelace formula suits anything already in coordinates, which is how mapping software computes parcel areas.

Where it gets used

Land. Irregular plots are split into triangles, each is measured, and the areas are added. This is triangulation, and it predates every other survey method.

Roofing. Gable ends and hip roof faces are triangles, and material is ordered by their area. A gable 8 m wide rising 3 m at the ridge covers ½ × 8 × 3 = 12 m².

Sailmaking. Sail area is quoted in square metres and drives a boat's rating. Most sails are triangles or close to it.

Computer graphics. Every 3D surface on a screen is broken into triangles, because three points always define a flat plane while four may not.

Equilateral and isosceles shortcuts

An equilateral triangle, all three sides equal, has area a²√3/4. A side of 10 gives 100 × 1.7320508 / 4 = 43.30 square units. Its height is a√3/2, which is 8.66 for that same triangle.

An isosceles triangle, two sides equal, splits down the middle into two right triangles. If the equal sides are b and the base is a, the height is √(b² − a²/4) and the area is half the base times that. A triangle with a 12 base and 10 equal sides has a height of √(100 − 36) = 8 and an area of 48.

Reading a triangle from coordinates

If the three corners are known as coordinates rather than side lengths, the shoelace formula is faster than measuring distances first. For corners (x₁,y₁), (x₂,y₂) and (x₃,y₃), the area is half the absolute value of x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂).

Take (0,0), (4,0) and (0,3). That gives ½ × |0(0−3) + 4(3−0) + 0(0−0)| = ½ × 12 = 6, matching the half-base-times-height answer for a right triangle with legs of 4 and 3.

The formula extends to any polygon, which is why mapping software uses it for parcel areas rather than breaking shapes into triangles by hand.

Triangles in a roof

Roof pitch is expressed as a rise over a run, such as 6 in 12, which means the roof climbs 6 units for every 12 horizontal. That gives a right triangle whose area is half the product of the two legs.

A gable 10 m wide has a horizontal run of 5 m each side. At a 6 in 12 pitch the rise is 2.5 m, so the gable triangle covers ½ × 10 × 2.5 = 12.5 m². The sloping rafter length comes from Pythagoras: √(25 + 6.25) = 5.59 m per side.

Roofers work in a rafter length rather than a plan width, because material follows the slope. That is why a roof always needs more covering than its footprint suggests, and the ratio between the two is fixed by the pitch.

Similar triangles and indirect measurement

Two triangles with the same angles have sides in the same ratio, however different their sizes. That single fact underpins most measurement of things too tall or too distant to reach.

Hold a ruler at arm's length, line it up against a distant building, and the ratio of ruler length to arm length equals the ratio of building height to its distance. If 10 cm of ruler at 60 cm from your eye covers a building 200 m away, the building is 200 × 10/60 = 33 m tall.

The same principle is behind rangefinders, the shadow method for pyramids, and the way a camera lens maps a scene onto a sensor. Areas scale with the square of the ratio, so a triangle twice the size of another has four times the area, not twice.

Common mistakes

Using a slanted side as the height. The height is perpendicular to the base. Using the sloping edge overstates the area.

Forgetting the half. Base times height gives the enclosing parallelogram, twice the triangle.

Feeding in impossible sides. Check that the two shorter sides add to more than the longest.

Mixing units. All three sides must share a unit before the arithmetic starts.