How To Calculate The Area Of A Triangle

Every triangle is half of a parallelogram. Draw a diagonal across a parallelogram and you cut it into two triangles of identical size, each using the same base and the same perpendicular height as the original figure. Because the parallelogram's area is base times height, each triangle takes exactly half of that. The factor of one half in A = 1/2 x base x height is therefore not a convention to memorise but a fact about how the shape splits. A base of 10 with a height of 6 sits inside a parallelogram of area 60, and the triangle takes 30.

Any of the three sides may serve as the base, and switching between them never changes the area. What changes is the height, because each base has its own perpendicular distance to the opposite corner and only that distance works. In a triangle with sides of 7, 8 and 9, taking the side of length 8 as the base gives a height of 6.70820393 and an area of 26.83281573; choosing either of the other sides gives a different height and the same 26.83281573. That freedom is practical rather than decorative, since the easiest side to measure is usually the one resting on the ground and the formula lets you begin there.

Height means perpendicular distance, and it is worth being pedantic about the word. It is the shortest distance from the opposite corner to the line containing the base, measured along a segment meeting that line at a right angle. In an obtuse triangle the foot of that perpendicular can land outside the triangle entirely, on the extension of the base rather than on the base itself, and nothing about that is an error. The formula still holds without adjustment. What must never happen is substituting the length of a slanted side, because a slanted side is always longer than the perpendicular distance it spans.

Units behave exactly as the formula demands, which is to say quadratically. Two lengths multiplied together produce square units, so measuring in metres gives square metres and measuring in centimetres gives square centimetres. Converting afterwards means squaring the conversion factor: one square metre is 10000 square centimetres, not 100. This is where otherwise correct arithmetic produces answers wrong by factors of a hundred or ten thousand. Writing the unit down straight after the number, before doing anything else with it, catches most of these slips at the moment they happen.

Heron's formula exists for the common case where three side lengths are known and no height is. Work out the semi-perimeter s = (a + b + c) / 2, then take A = sqrt(s(s - a)(s - b)(s - c)). For sides of 7, 8 and 9 the semi-perimeter is 12, the product inside the root is 12 x 5 x 4 x 3 = 720, and the square root is 26.83281573. The perimeter of 24 is a separate quantity that never enters the formula directly. Because s appears in three of the four factors, the result is more sensitive to an error in s than the base-and-height route would ever be.

Three lengths only describe a triangle when each one is shorter than the sum of the other two. Sides of 1, 2 and 10 fail that test, and no such triangle exists whatever a formula might return. Under Heron's formula the failure surfaces as a negative value inside the square root, which is why a careful implementation clamps the product at zero before taking the root. Satisfying the inequality is not quite sufficient either: an extremely flat triangle has a very small height and a very small area, so the computed figure carries far less precision than the digits suggest.

When two sides and the angle between them are known, hunting for a height is unnecessary. Dropping a perpendicular shows that the height to side a is b x sin C, and substituting that into the base-and-height formula gives A = 1/2 x a x b x sin C. This is the same relation wearing different clothes, and it is the natural choice for surveying work where a corner angle is easier to measure than a perpendicular distance. It also makes visible something the other forms hide: with two side lengths fixed, the area is largest when the angle between them is a right angle.

Two special shapes repay memorising, because each removes a step. In a right-angled triangle the legs are already perpendicular, so one leg is the base and the other is the height: legs of 3 and 4 give an area of 6, with a hypotenuse of 5. An equilateral triangle of side a has area a^2 x sqrt(3) / 4, so a side of 6 gives 36 x 1.73205081 / 4 = 15.58845727, and its own height is 5.19615242. Recognising these saves the effort of constructing a height that the shape has effectively handed you already.

Precision should follow the measurement rather than the calculator. If the base and height were read to the nearest centimetre, reporting eight decimal places claims an accuracy that was never there. Carry full precision through the intermediate steps so that rounding does not accumulate, then round once at the end to something the original measurement can defend. The advice matters more for Heron's formula than for the base-and-height route, since the semi-perimeter is subtracted from each side and a small absolute error there becomes a large relative one in the factors that follow.