Area of a triangle: Vertices (Part 7)

Suppose that the vertices of a triangle are (1,2), (2,5), and (3,1). What is the area of the triangle?

At first blush, this doesn’t fall under any of the categories of SSS, SAS, or ASA. And we certainly aren’t given a base b and a matching height h. The Pythagorean theorem could be used to determine the lengths of the three sides so that Heron’s formula could be used, but that would be extremely painful to do.

Fortunately, there’s another way to find the area of a triangle that directly uses the coordinates of the triangle. It turns out that the area of the triangle is equal to the absolute value of

\displaystyle \frac{1}{2} \left| \begin{array}{ccc} 1 & 2 & 1 \\ 2 & 5 & 1 \\ 3 & 1 & 1 \end{array} \right|

Notice that the first two columns contain the coordinates of the three vertices, while the third column is just padded with 1s. Calculating, we find that the area is

\left| \displaystyle \frac{1}{2} \left( 5 + 6 + 2 - 15 - 4 - 1 \right) \right| = \left| \displaystyle -\frac{7}{2} \right| = \displaystyle \frac{7}{2}

In other words, direct use of the vertices is, in this case, a lot easier than the standard SSS, SAS, or ASA formulas.

A (perhaps) surprising consequence of this formula is that the area of any triangle with integer coordinates must either be an integer or else a half-integer. We’ll see this again when we consider Pick’s theorem in tomorrow’s post.

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There is another way to solve this problem by considering the three vertices as points in \mathbb{R}^3. The vector from (1,2,0) to (2,5,0) is \langle 1,3,0 \rangle, while the vector from (1,2,0) to (3,1,0) is \langle 2,-1,0 \rangle. Therefore, the area of the triangle is one-half the length of the cross-product of these two vectors. Recall that the cross-product of the two vectors is

\langle 1,3,0 \rangle \times \langle 2,-1,0 \rangle = \left| \begin{array}{ccc} {\bf i} & {\bf j} & {\bf k} \\ 1 & 3 & 0 \\ 2 & -1 & 0 \end{array} \right|

\langle 1,3,0 \rangle \times \langle 2,-1,0 \rangle = -7{\bf k}

So the length of the cross-product is clearly 7, so that the area of the triangle is (again) \displaystyle \frac{7}{2}.

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The above technique works for any triangle in \mathbb{R}^3. For example, if we consider a triangle in three-dimensional space with corners at (1,2,3), (4,3,0), and (6,1,9), the area of the triangle may be found by “subtracting” the coordinates to find two vectors along the sides of the triangle and then finding the cross-product of those two vectors.

Furthermore, determinants may be used to find the volume of a tetrahedron in \mathbb{R}^3. Suppose that we now consider the tetrahedron with corners at (1,2,3), (4,3,0), (6,1,9), and (2,5,2). Let’s consider (1,2,3) as the “starting” point and subtract these coordinates from those of the other three points. We then get the three vectors

\langle 3,2,-3 \rangle, \langle 5,-1,6 \rangle, and \langle 1,3,-1 \rangle

One-third of the absolute value of the determinant of these three vectors will be the volume of the tetrahedron.

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This post has revolved around one central idea: a determinant represents an area or a volume. While this particular post has primarily concerned triangles and tetrahedra, I should also mention that determinants are similarly used (without the factors of 1/2 and 1/3) for finding the areas of parallelograms and the volumes of parallelepipeds.

This central idea is also the basis behind an important technique taught in multivariable calculus: integration in polar coordinates and in spherical coordinates.
In two dimensions, the formulas for conversion from polar to rectangular coordinates are

x = r \cos \theta and y = r \sin \theta

Therefore, using the Jacobian, the “infinitesimal area element” used for integrating is

dx dy = \left| \begin{array}{cc} \partial x/\partial r & \partial y/\partial r \\ \partial x/\partial \theta & \partial y/\partial \theta \end{array} \right| dr d\theta

dx dy = \left| \begin{array}{cc} \cos \theta & \sin \theta \\ -r \sin \theta & r \cos \theta \end{array} \right| dr d\theta

dx dy = (r \cos^2 \theta + r \sin^2 \theta) dr d\theta

dx dy = r dr d\theta

Similarly, using a 3 \times 3 determinant, the conversion dx dy dz = r^2 \sin \phi dr d\theta d\phi for spherical coordinates can be obtained.