Intereting Posts

Arranging letters with two letters not next to each other
Relationship Between Ratio Test and Power Series Radius of Convergence
Lottery odds calculated in your head, or pen and paper.
Symmetric random walk and the distribution of the visits of some state
Generalized Riemann Integral: Bounded Nonexample?
A Borel set whose projection onto the first coordinate is not a Borel set
How to prove Mandelbrot set is simply connected?
Intuition behind proof of bounded convergence theorem in Stein-Shakarchi
Prove $\lim_{n\to\infty}x_n=2$ Given $\lim_{n \to \infty} x_n^{x_n} = 4$
The Abelianization of $\langle x, a \mid a^2x=xa\rangle$
limit of a recursively defined function
A net version of dominated convergence?
A limit wrong using Wolfram Alpha
Prob. 9, Sec. 4.3 in Kreyszig's Functional Analysis Book: Proof of the Hahn Banach Theorem without Zorn's Lemma
Borel $\sigma$-Algebra definition.

What is the simplest way to find out the area of a triangle if the coordinates of the three vertices are given in $x$-$y$ plane?

One approach is to find the length of each side from the coordinates given and then apply *Heron’s formula*. Is this the best way possible?

Is it possible to compare the area of triangles with their coordinates provided without actually calculating side lengths?

- Does anyone know of any open source software for drawing/calculating the area of intersection of different shapes?
- Area of ascending regions on implicit plot - $\cos(y^y) = \sin(x)$
- What is “Squaring the Circle”
- Trisect a quadrilateral into a $9$-grid; the middle has $1/9$ the area
- find maximum area
- Triangles area question

- Area of a parallelogram, vertices $(-1,-1), (4,1), (5,3), (10,5)$.
- Consider a right angled $\triangle PQR$ right angled at $P$ i.e ($\angle QPR=90°$) with side $PR=4$ and area$=6$.
- Getting a transformation matrix from a normal vector
- Pullback metric, coordinate vector fields..
- Area under parabola using geometry
- invariance of cross product under coordinates rotation
- under what conditions can orthogonal vector fields make curvilinear coordinate system?
- The Area of an Irregular Hexagon
- Find X location using 3 known (X,Y) location using trilateration
- What is the chance that an $n$-gon whose vertices lie randomly on a circle's circumference covers a majority of the circle's area?

What you are looking for is the shoelace formula:

\begin{align*}

\text{Area}

&= \frac12 \big| (x_A – x_C) (y_B – y_A) – (x_A – x_B) (y_C – y_A) \big|\\

&= \frac12 \big| x_A y_B + x_B y_C + x_C y_A – x_A y_C – x_C y_B – x_B y_A \big|\\

&= \frac12 \big|\det \begin{bmatrix}

x_A & x_B & x_C \\

y_A & y_B & y_C \\

1 & 1 & 1

\end{bmatrix}\big|

\end{align*}

The last version tells you how to generalize the formula to higher dimensions.

Heron’s formula is inefficient; there is in fact a direct formula. If the triangle has one vertex at the origin, and the other two vertices are $(a,b)$ and $(c,d)$, the formula for its area is

$$

A = \frac{\left| ad – bc \right|}{2}

$$

To get a formula where the vertices can be anywhere, just subtract the coordinates of the third vertex from the coordinates of the other two (translating the triangle) and then use the above formula.

The simplest way to remember how to calculate is by taking $\frac{1}{2}$ the value of the determinant of the matrix

$$

\begin{bmatrix}

1 & 1 & 1 \\

x_1 & x_2 & x_3 \\

y_1 & y_2 & y_3

\end{bmatrix}

$$

You know that **AB x AC** is a vector perpendicular to the plan ABC such that |**AB x AC**|= Area of the parallelogram ABA’C. Thus this area is equal to ½ |AB x AC|.

From **AB**= $(x_2 -x_1, y_2-y_1)$; **AC**= $(x_3-x_1, y_3-y_1)$, we deduce then

Area of $\Delta ABC$ = $\frac12$$[(x_2-x_1)(y_3-y_1)- (x_3-x_1)(y_2-y_1)]$

if $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ are the vertices of a triangle then its area is given by

$$|\frac12(x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2))|$$

The area of a triangle $P(x_1, y_1)$, $Q(x_2, y_2)$ and $R(x_3, y_3)$ is given by

$$\triangle= \left|\frac{1}{2}(x_{1} (y_{2} – y_{3}) + x_{2} (y_{3} – y_{1}) + x_{3} (y_{1} – y_{2}))\right| $$

If the area of triangle is zero, it means the points are collinear.

If we code this in `Python3`

, it will look like:

```
def triangle_area(x1, y1, x2, y2, x3, y3):
return abs(0.5*(x1*(y1-y2)+x2*(y3-y1)+x3*(y1-y2)))
```

The area $A$ of the triangle two of whose vertices lie on the axes, with coordinates $(a, 0)$, $(0, b)$, and a third vertex $(c, d)$ is obtained from previous formula by a mere horizontal axis shift of -a units as

$$A = \frac{|-ad + b(a – c)|}{2}$$

For fun, I’ll just throw out the ** really long** way that I learned in 3rd grade, only because it hasn’t been submitted. I

- Determine the distance between two of the three points, say $ \big( x_{1}, y_{1} \big) $ and $ \big( x_{2}, y_{2} \big) $.

$d = \sqrt{ \big(x_{2} – x_{1}\big) ^{2} + \big(y_{2} – y_{1}\big) ^{2} } $ - Determine the slope $m = \frac{y_{2} – y_{1}}{x_{2} – x_{1}} $ and y-intercept, $a = y_{1} – \big(m \times x_{1}\big)$, of the line between $ \big( x_{1}, y_{1} \big) $ and $ \big( x_{2}, y_{2} \big) $.
- Determine the slope of the a line perpendicular the line from $ \big( x_{1}, y_{1} \big) $ and $ \big( x_{2}, y_{2} \big) $, which is the negative reciprocal of the first slope. $n = \frac{-1}{m} $
- Determine the equation of the line parallel to this second line, that passes through the third point $ \big( x_{3}, y_{3} \big)$, by finding the y-intercept in $y = n*x+b$, since you already have the slope and a point on the line. $ y_{3} – \big(n \times x_{3}\big)=b$.
- Determine where this new line intersects the line between $ \big( x_{1}, y_{1} \big) $ and $ \big( x_{2}, y_{2} \big) $, by solving the system of equations of the new line and the original line: $y = m*x+b$ and $y = n*x+b$. Call this point $ \big( x_{4}, y_{4} \big) $. I won’t write this out, I’ll leave it as an “exercise for the reader”.
- Determine the distance between $ \big( x_{3}, y_{3} \big) $ and the new point$ \big( x_{4}, y_{4} \big) $.

$c = \sqrt{ \big(x_{4} – x_{3}\big) ^{2} + \big(y_{4} – y_{3}\big) ^{2} } $ - If $d$ is the base of the triangle, and $c$ the height, the area is $A = \frac{1}{2} c*d$.
- Realize you’ve spent several minutes solving a trivial problem… cry silently.

- Calculating Christoffel symbols using variational geodesic equation
- Proving that $\frac{\pi}{4}$$=1-\frac{\eta(1)}{2}+\frac{\eta(2)}{4}-\frac{\eta(3)}{8}+\cdots$
- Bounded index of nilpotency
- When do equations represent the same curve?
- Prove that $f=x^4-4x^2+16\in\mathbb{Q}$ is irreducible
- Assume that $(\text{X}, T)$ is compact and Hausdorff. Prove that a comparable but different topological space $(\text{X},T')$ is not.
- Monotone class theorem vs Dynkin $\pi-\lambda$ theorem
- Numerical methods book
- evaluating the limit $ \lim_{p \rightarrow (0,0) } \frac{x^2y^2}{x^2 + y^2} $
- irreducibility of polynomials with integer coefficients
- weak convergence in $L^p$ plus convergence of norm implies strong convergence
- Explain a surprisingly simple optimization result
- If $\lim_{n\to\infty}a_n=a$ and $a_n>0$ for all $n$, then we have $ \lim_{n\to\infty}\sqrt{a_1a_2\cdots a_n}=a $?
- $f:\mathbb R\to\mathbb R$ continuous function. Which of the following sets can not be image of $(0,1]$ under $f$?
- Is there a simple proof of Borsuk-Ulam, given Brouwer?