# Area of triangle

1. Jun 3, 2014

1. The problem statement, all variables and given/known data
Find the area of triangle formed by the points $A(5,2)$ , $B(4,7)$ , $C(7,-4)$

2. Relevant equations
Nah

3. The attempt at a solution
Is there any better way than finding the angle between lines and their lengths and then the area?

2. Jun 3, 2014

### SteamKing

Staff Emeritus
That method sounds pretty tedious.

You can treat the triangle as a polygon with three sides. This article has a neat formula for finding the areas of polygons knowing only the coordinates of the vertices:

http://en.wikipedia.org/wiki/Polygon

Some words of caution:
1. the polygon must be traversed counterclockwise w.r.t. the origin to accumulate positive area.
2. the polygon must be closed, which implies that P1 = Pn, where Pk represents the coordinates of the kth vertex of the polygon.

3. Jun 3, 2014

Is it this formula?

4. Jun 3, 2014

### BOAS

You could find the area via the vector cross product.

$A = \frac{1}{2}|\vec{AB} \times \vec{AC}|$

5. Jun 3, 2014

### Fightfish

6. Jun 3, 2014

Oh. I haven't studied vector products yet. It's $\frac{1}{2} AB \times AC \times \sin(\theta)$ right?

Last edited: Jun 3, 2014
7. Jun 3, 2014

### BOAS

$\frac{1}{2}|\vec{AB} \times \vec{AC}|$ means to take the vector product of the two vectors, find the magnitude of the resultant vector (which gives the area of a parallelogram) and then multiply by 0.5 to get the area of a triangle.

In order to take the vector product you use a 'determinant' - Though you don't need to have studied matrices in order to perform this operation.

http://mathworld.wolfram.com/CrossProduct.html

8. Jun 3, 2014

### SteamKing

Staff Emeritus
Yes, that's the one.

9. Jun 3, 2014

### SammyS

Staff Emeritus
Yes, $\ \frac{1}{2}|\vec{AB} \times \vec{AC}|=\frac{1}{2}\cdot |AB|\cdot |AC|\cdot \sin(\theta)\ .$

But doing the vector product using components, as suggested by BOAS eliminates the need to determine θ. In fact doing the vector product, or scalar product for that matter, via components allows you to determine θ.

10. Jun 3, 2014

I will have to study advanced vectors then. When will it be covered?

11. Jun 3, 2014

### Ray Vickson

Alternatively, you could use Heron's formula, which states that for a triangle with sides $a, b, c$ the area $A$ is given by
$$A = \sqrt{s(s-a)(s-b)(s-c)},\: \text{ where } \: s = \frac{a+b+c}{2}$$
See, eg., http://en.wikipedia.org/wiki/Heron's_formula .

12. Jun 3, 2014

### Saitama

Do you know about evaluating determinants? If so, there's a formula you might have seen.

Or maybe, does the following look familiar?
$$\frac{1}{2}\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|$$

13. Jun 3, 2014

Is that some kind of determinant? Can you write that in the matrix form(If it has a matrix form)?

14. Jun 3, 2014

### Saitama

I am not sure what you ask here. Did you comment on the formula I wrote?

The formula in the determinant form is:
$$\frac{1}{2} \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end{vmatrix}$$

15. Jun 3, 2014

OMG, what is that? I have only studied to find the determinant of a 2x2 matrix
$$\begin{matrix} a && b \\ c && d\\ \end{matrix}$$

16. Jun 3, 2014

### Saitama

Evaluating 3X3 determinants is very similar to 2X2 ones. You should look it up.

17. Jun 3, 2014

Why do we need a 3x3 matrix for calculating the area?

18. Jun 3, 2014

### verty

If you know about direction cosines, you can just use the standard area formula without any of this fancy trickery.

Oh sorry, the question was, is there a better way? Not really, this is a very good way. But determinants are equally good, and a 2x2 can be used. There was a recent thread about this: here.

All I can say is, learn more linear algebra to use that method.

Last edited: Jun 3, 2014
19. Jun 3, 2014

### verty

I'll add, these are linear algebra topics, usually that is not precalculus math and therefore I recommend that people use the standard formula, or of course they can learn in advance to use something more advanced.