 | Triangle: Encyclopedia II - Triangle - Computing the area of a triangle
Triangle - Computing the area of a triangle
Calculating the area of a triangle is an elementary problem encountered often in many different situations. Various approaches exist, depending on what is known about the triangle. What follows is a selection of frequently used formulae for the area of a triangle.
Triangle - Using geometry
The area S of a triangle is S = ½bh, where b is the length of any side of the triangle (the base) and h (the altitude) is the perpendicular distance between the base and the vertex not on the base. This can be shown with the following geometric construction.
To find the area of a given triangle (green), first make an exact copy of the triangle (blue), rotate it 180°, and join it to the given triangle along one side to obtain a parallelogram. Cut off a part and join it at the other side of the parallelogram to form a rectangle. Because the area of the rectangle is bh, the area of the given triangle must be ½bh.
Triangle - Using vectors
The area of a parallelogram can also be calculated by the use of vectors. If AB and AC are vectors pointing from A to B and from A to C, respectively, the area of parallelogram ABDC is |AB × AC|, the magnitude of the cross product of vectors AB and AC. |AB × AC| is also equal to |h × AC|, where h represents the altitude h as a vector.
The area of triangle ABC is half of this, or S = ½|AB × AC|.
Triangle - Using trigonometry
The altitude of a triangle can be found through an application of trigonometry. Using the labelling as in the image on the left, the altitude is h = a sin γ. Substituting this in the formula S = ½bh derived above, the area of the triangle can be expressed as S = ½ab sin γ.
It is of course no coincidence that the area of a parallelogram is ab sin γ.
Triangle - Using coordinates
If vertex A is located at the origin (0, 0) of a Cartesian coordinate system and the coordinates of the other two vertices are given by B = (x1, y1) and C = (x2, y2), then the area S can be computed as 1/2 times the absolute value of the determinant
or S = ½ |x1y2 − x2y1|.
For three general vertices the equation is:
In three dimensions the area of a general triangle {A = (x1, y1, z1), B = (x2, y2, z2) and C = (x3, y3, z3)} is the 'pytagorean' sum of the area's of the respective projections on the three principal planes (i.e. x=0, y=0 and z=0):
Triangle - Using Heron's formula
Yet another way to compute S is Heron's formula:
where s = ½ (a + b + c) is the semiperimeter, or one half of the triangle's perimeter.
Other related archives300 BCE, Cartesian coordinate system, Ceva's theorem, Elements, Euclid, Euler's line, Feuerbach point, Heron's formula, Menelaus' theorem, Pythagorean theorem, Thales' theorem, absolute value, acute angles, altitude, angle, angle bisector, angles, area, center of gravity, centroid, circle, circumcenter, circumcircle, collinear, concurrent, cosine, cross product, degrees, determinant, dimensional, excircles, geometry, hyperbolic geometry, hyperbolic triangles, incenter, incircle, law of cosines, law of sines, line segments, median, nine-point circle, obtuse angle, orthocenter, orthocentric system, parallelogram, perpendicular bisector, plane, polygon, polytope, proportional, regular polygon, right angle, similar, simplex, sine, special right triangles, spherical geometry, spherical triangles, straight, symmedian, symmedian point, trigonometric functions, trigonometry, vectors, vertices
 Adapted from the Wikipedia article "Computing the area of a triangle", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
