Triangle Area Calculator
Calculate the area of a triangle by entering base and height. Fast and clear result with A = (base x height) / 2.
Triangle area: five formulas for five situations
Half base times height only works when you actually know the height, and in surveying, land measurement, and most real problems you do not. This page covers the alternatives - Heron, the sine rule, and the coordinate method - and when each one is the right tool.
Formula
A = (base x height) / 2
The height must be the perpendicular distance from the base to the opposite vertex, not the length of a slanted side.
Perpendicular height is not a side length
This is the error that dominates. In a right-angled triangle the two legs happen to serve as base and height, which teaches people that a side is a height. In any other triangle it is not.
For an obtuse triangle the foot of the perpendicular falls outside the base entirely, so the height is measured to an extension of the base line. The formula still holds; the geometry just looks unfamiliar.
A triangle with sides 5, 6, and 7 has no side that is a height. Using 5 and 6 as base and height gives 15, while the true area is 14.7 - close enough to look plausible, which is what makes the error dangerous.
When you know three sides: Heron's formula
Heron's formula needs only the side lengths, which is exactly what a tape measure gives you. Compute the semi-perimeter s = (a + b + c) / 2, then A = sqrt(s(s-a)(s-b)(s-c)).
For sides 5, 6, 7: s = 9, and A = sqrt(9 x 4 x 3 x 2) = sqrt(216) = 14.70.
A useful side effect is validation. If any of the three bracketed terms is negative or zero, the sides cannot form a triangle - the triangle inequality has been violated, meaning one side is longer than the other two combined.
Two sides and the angle between them
When you have two sides and their included angle, A = (1/2) x a x b x sin(C). This is the standard tool in surveying and navigation, because angles are easier to measure accurately over distance than perpendicular offsets.
The angle must be the one between the two sides you are using. Using a different angle gives a number that is silently wrong. Note also that sin(C) equals sin(180 - C), so this formula alone cannot distinguish an acute triangle from its obtuse counterpart.
From coordinates: the shoelace method
Given three vertices, the area is |x1(y2-y3) + x2(y3-y1) + x3(y1-y2)| / 2. This is the shoelace formula restricted to three points, and it is how GIS software computes parcel areas.
For vertices (0,0), (4,0), and (1,3): |0(0-3) + 4(3-0) + 1(0-0)| / 2 = 12 / 2 = 6. It requires no trigonometry and generalises directly to any polygon.
Worked example
A triangular garden plot measures 18.4 m, 22.7 m, and 15.9 m along its boundaries. No height is available and the corners are not accessible for angle measurement.
Semi-perimeter: s = (18.4 + 22.7 + 15.9) / 2 = 28.5. Then A = sqrt(28.5 x 10.1 x 5.8 x 12.6) = sqrt(21,033) = 145.0 m^2.
Cross-check with the triangle inequality: 18.4 + 15.9 = 34.3, comfortably more than 22.7, so the measurements are consistent.
Choosing a triangle area formula
| What you know | Formula | Typical use |
|---|---|---|
| Base and perpendicular height | A = b x h / 2 | Drawings, right triangles |
| Three sides | Heron: sqrt(s(s-a)(s-b)(s-c)) | Land and site measurement |
| Two sides and included angle | A = a x b x sin(C) / 2 | Surveying, navigation |
| Three vertex coordinates | Shoelace formula | GIS, CAD, mapping |
| Equilateral, side a | A = sqrt(3) x a^2 / 4 | Structural and tiling work |
Frequently Asked Questions
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References
Last reviewed: 2026-08-07. This page is informational. For legal, medical, tax, or financial decisions, confirm the result with a qualified professional.