Triangle Area Calculator

Background.

Half the base times the height is the formula everyone remembers, and it is exact for every triangle there is. The trouble is the height. It has to be measured at right angles to the base, from the base line up to the opposite corner, and on a real object, such as a gable end, a garden bed or a plot of land, that line is rarely drawn anywhere you can put a tape on it. Measure a sloping side instead and the area comes out too large.

That is why there are three routes to the same number, and you can choose whichever matches what you actually measured. If you have a base and a true perpendicular height, A = ½ × b × h. If you have the three side lengths and nothing else, Heron's formula gets the area from the sides alone, with no angle and no height; it is named after Heron of Alexandria, a first-century mathematician and engineer who set it out in his Metrica. If you know two sides and the angle where they meet, perhaps from a drawing or a protractor, A = ½ × a × b × sin C.

The three are not rival approximations. Each is the same quantity reached from different measurements, so one physical triangle gives one area whichever way you go, and working it out a second way is a good check on a doubtful measurement.

Two kinds of input have no triangle behind them and are refused. Three sides where one is as long as the other two together, such as 2, 3 and 5, lie flat in a line and enclose nothing. An angle of 0° or 180° between two sides does the same. Lengths can be in any unit as long as all of them use the same one, and the area comes out in that unit squared.

What is triangle area calculator?

The area of a triangle is the flat region enclosed by its three sides, measured in square units. Any side can serve as the base; the height belonging to that base is the perpendicular distance from the base's line to the opposite vertex. In an obtuse triangle that perpendicular can land outside the base itself, on the base line extended. A triangle is exactly half of a parallelogram on the same base and between the same parallels, which is Euclid's Elements, Book I, Proposition 41, and is where the half in ½ × base × height comes from.

How to use this calculator.

1. 01Choose the mode that matches your measurements: Base × Height, SSS for three sides, or SAS for two sides and the angle between them. Only the fields for that mode are used.
2. 02In Base × Height, enter a side as the base and the perpendicular distance from it to the opposite corner as the height, not the length of a sloping side.
3. 03In SSS, enter sides a, b and c in any order. If one side is equal to or longer than the other two combined, no triangle exists and the input is refused.
4. 04In SAS, enter sides a and b and the angle C between them, in degrees, greater than 0 and less than 180. It must be the angle where those two sides meet, not one of the other two.
5. 05Read the area in the square of your length unit, and the method label confirming which formula was applied.

The formula.

Base × Height: A = ½ × b × h. Copy the triangle, turn the copy through 180° and fit it against one side, and the two make a parallelogram with area b × h; the triangle is half of it.

Heron's formula: A = √(s(s − a)(s − b)(s − c)), where s = (a + b + c) ÷ 2 is the semiperimeter, half the distance around the triangle. Each factor s − a, s − b and s − c is positive exactly when that side is shorter than the other two together, which is the triangle inequality, so the expression under the root is positive for every real triangle. One way to derive it is to find an angle from the three sides with the law of cosines and substitute it into the SAS formula; the algebra collapses into Heron's tidy product.

SAS: A = ½ × a × b × sin C. Taking a as the base, the height from the far end of b is b × sin C, so the formula is base times height again, with the height worked out by trigonometry. At C = 90°, sin C = 1 and the formula becomes half the product of the two legs, the right-triangle case. Because sin C = sin(180° − C), an angle of 30° and one of 150° between the same two sides give the same area.

The arithmetic is exact; any error comes from the measurements. Heron's formula is sensitive to them when the triangle is very flat, because one of the factors s − a, s − b or s − c is then close to zero.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Euclid, Elements, Book I, Proposition 41 — a parallelogram on the same base and between the same parallels as a triangle is double the triangle; the classical basis of A = (1/2)·base·height (David E. Joyce edition, Clark University).
2. [2]MacTutor History of Mathematics Archive, University of St Andrews — "Heron of Alexandria," covering the Metrica and the three-sides area formula now known as Heron's formula.
3. [3]NIST Digital Library of Mathematical Functions, §4.14 — definitions of the circular (trigonometric) functions, the basis of the (1/2)·a·b·sin C formula used in SAS mode.
4. [4]Coxeter, H. S. M. Introduction to Geometry, 2nd edition. Wiley, 1969 — classic reference textbook covering triangle area formulas, Heron's formula, and the triangle inequality.