What is the simplest formula for the area of a triangle?

Area = (1/2) x base x height, where the height is the perpendicular distance from the base to the opposite vertex. This works for any triangle type as long as you measure the perpendicular (not slant) height.

How does Heron''s formula work?

Given three side lengths a, b, c, compute the semi-perimeter s = (a + b + c) / 2. Then Area = sqrt(s(s-a)(s-b)(s-c)). No angles are needed — just the three side lengths.

When should I use the SAS (two sides and included angle) method?

Use Area = (1/2) x a x b x sin(C) when you know two side lengths and the angle between them. This is common in surveying and navigation where measuring an angle is easier than dropping a perpendicular height.

What is the Shoelace formula and when do I need it?

The Shoelace (or Surveyor's) formula computes the area of any polygon from its vertex coordinates: Area = |x1(y2-y3) + x2(y3-y1) + x3(y1-y2)| / 2 for a triangle. It is the standard method in GIS, CAD, and computer graphics where points come from a coordinate system.

Does the calculator work for obtuse and right triangles?

Yes. All four methods work for acute, right, and obtuse triangles. Heron''s formula and the coordinate method make no assumption about angle type. The SAS method handles obtuse included angles (0 to 180 degrees exclusive) correctly because sin(C) remains positive throughout that range.

Why does my triangle fail the inequality check?

A valid triangle requires every side to be shorter than the sum of the other two (a + b > c, a + c > b, b + c > a). If this is violated, the three lengths cannot close into a triangle — for example, sides 1, 2, and 10 would need the short sides to stretch impossibly far to meet the long one.

Triangle Area Calculator

A triangle area calculator that supports four different input methods — so you can always solve from the values you actually have, without rearranging formulas by hand. Whether you know the base and height, all three side lengths, two sides and their included angle, or the Cartesian coordinates of all three vertices, the tool computes the area instantly and shows you every arithmetic step.

Four methods, one place

Triangles appear in architecture, surveying, engineering, and everyday geometry problems, but the measurements available vary by situation. This calculator covers every common case.

Base and height is the fastest method when you can measure or are given the perpendicular height. The formula Area = (1/2) x base x height is the most widely taught and holds for any triangle orientation.

Heron’s formula requires only the three side lengths — no angles, no height. It is ideal when you have measured all three sides of a triangular plot of land or a structural member but have no convenient right angle to exploit. The semi-perimeter s = (a + b + c) / 2 is computed first, then Area = sqrt(s(s-a)(s-b)(s-c)).

Two sides and the included angle (SAS) uses Area = (1/2) x a x b x sin(C), where C is the angle between the two known sides. This is the standard surveying formula: measuring an angle is often easier than finding a perpendicular height, and the sine function handles any angle from just above 0 degrees to just below 180 degrees.

Coordinate vertices (Shoelace theorem) takes the (x, y) positions of all three corners and applies the signed area formula Area = |x1(y2-y3) + x2(y3-y1) + x3(y1-y2)| / 2. This is the method used in GIS software, CAD packages, and computer graphics to compute polygon areas directly from point data.

SVG diagram and step-by-step working

Each method draws a proportional SVG outline of the computed triangle labelled A, B, C so you can sanity-check the shape. Clicking Show working expands a step-by-step arithmetic trace — useful for checking homework, verifying a formula, or explaining the result to someone else. The Copy button copies the full working to your clipboard in plain text.

Worked example — Heron’s formula

A triangular garden has sides of 9 m, 12 m, and 15 m.

Semi-perimeter: s = (9 + 12 + 15) / 2 = 18
s - a = 9, s - b = 6, s - c = 3
Product: 18 x 9 x 6 x 3 = 2916
Area = sqrt(2916) = 54 m

Notice that 9-12-15 is a 3-4-5 right triangle scaled by 3, so the quick check is (1/2) x 9 x 12 = 54 m — matching exactly.

Method	Inputs	Area
Base and height	base 10, height 6	30 sq units
Heron’s formula	sides 9, 12, 15	54 sq units
SAS	sides 8, 5, angle 60 deg	17.32 sq units
Coordinates	(0,0), (6,0), (3,4)	12 sq units

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