Shoelace formula

The formula can be represented by the expression

${\displaystyle {\begin{aligned}\mathbf {A} &={1 \over 2}\left|\left(\sum _{i=1}^{n-1}x_{i}y_{i+1}\right)+x_{n}y_{1}-\left(\sum _{i=1}^{n-1}x_{i+1}y_{i}\right)-x_{1}y_{n}\right|\\[4pt]&={1 \over 2}|x_{1}y_{2}+x_{2}y_{3}+\cdots +x_{n-1}y_{n}+x_{n}y_{1}-x_{2}y_{1}-x_{3}y_{2}-\cdots -x_{n}y_{n-1}-x_{1}y_{n}|\end{aligned}}}$

where

• A is the area of the polygon,
• n is the number of sides of the polygon, and
• (x_i, y_i), i = 1, 2, ..., n are the ordered vertices (or "corners") of the polygon.

Alternatively[3][7][8]

${\displaystyle {\begin{aligned}\mathbf {A} &={1 \over 2}\left|\sum _{i=1}^{n}x_{i}(y_{i+1}-y_{i-1})\right|={1 \over 2}\left|\sum _{i=1}^{n}y_{i}(x_{i+1}-x_{i-1})\right|={1 \over 2}\left|\sum _{i=1}^{n}(x_{i}y_{i+1}-x_{i+1}y_{i})\right|\\[5pt]&={1 \over 2}\left|\sum _{i=1}^{n}(x_{i+1}+x_{i})(y_{i+1}-y_{i})\right|={1 \over 2}\left|\sum _{i=1}^{n}\det {\begin{pmatrix}x_{i}&x_{i+1}\\y_{i}&y_{i+1}\end{pmatrix}}\right|\end{aligned}}}$

where x_n+1 = x₁ and x₀ = x_n, as well as y_n+1 = y₁ and y₀ = y_n.

If the points are labeled sequentially in the counterclockwise direction, then the sum of the above determinants is positive and the absolute value signs can be omitted;[1] if they are labeled in the clockwise direction, the sum of the determinants will be negative. This is because the formula can be viewed as a special case of Green's Theorem.

A particularly concise statement of the formula can be given in terms of the exterior algebra. If ${\displaystyle v_{1},\dots ,v_{n}}$ are the consecutive vertices of the polygon (regarded as vectors in the Cartesian plane) then

${\displaystyle A={\frac {1}{2}}\cdot \left|\sum _{i=1}^{n}v_{i}\wedge v_{i+1}\right|.}$

The user must know the points of the polygon in a Cartesian plane. For example, take a triangle with coordinates {(2, 1), (4, 5), (7, 8)}. Take the first x-coordinate and multiply it by the second y-value, then take the second x-coordinate and multiply it by the third y-value, and repeat as many times until it is done for all wanted points. This can be defined by this formula:[9]

${\displaystyle \mathbf {A} _{\text{tri.}}={1 \over 2}|x_{1}y_{2}+x_{2}y_{3}+x_{3}y_{1}-x_{2}y_{1}-x_{3}y_{2}-x_{1}y_{3}|}$

for x_i and y_i representing each respective coordinate. This formula is just the expansion of those given above for the case n = 3. Using it, one can find that the area of the triangle equals one half of the absolute value of 10 + 32 + 7 − 4 − 35 − 16, which equals 3. The number of variables depends on the number of sides of the polygon. For example, a pentagon will be defined up to x₅ and y₅:

${\displaystyle \mathbf {A} _{\text{pent.}}={1 \over 2}|x_{1}y_{2}+x_{2}y_{3}+x_{3}y_{4}+x_{4}y_{5}+x_{5}y_{1}-x_{2}y_{1}-x_{3}y_{2}-x_{4}y_{3}-x_{5}y_{4}-x_{1}y_{5}|}$

A quadrilateral will be defined up to x₄ and y₄:

${\displaystyle \mathbf {A} _{\text{quad.}}={1 \over 2}|x_{1}y_{2}+x_{2}y_{3}+x_{3}y_{4}+x_{4}y_{1}-x_{2}y_{1}-x_{3}y_{2}-x_{4}y_{3}-x_{1}y_{4}|}$

Consider the polygon defined by the points (3,4), (5,11), (12,8), (9,5) and (5,6), and illustrated in the following diagram:

Figure of this example

The area of this polygon is:

${\displaystyle {\begin{aligned}\mathbf {A} &={1 \over 2}\,{\big |}3\times 11+5\times 8+12\times 5+9\times 6+5\times 4\\&{}\qquad {}-4\times 5-11\times 12-8\times 9-5\times 5-6\times 3{\big |}\\[10pt]&={60 \over 2}=30\end{aligned}}}$

The reason this formula is called the shoelace formula is because of a common method used to evaluate it. This method uses matrices. As an example, choose the triangle with vertices (2,4), (3,−8), and (1,2). Then construct the following matrix by “walking around” the triangle and ending with the initial point.[10]

${\displaystyle {\begin{bmatrix}2&4\\3&-8\\1&2\\2&4\end{bmatrix}}}$

First, draw diagonal down and to the right slashes (as shown below),

and multiply the two numbers connected by each slash, then add all the products: (2 × −8) + (3 × 2) + (1 × 4) = −6. Do the same thing with slashes diagonal down and to the left (shown below with downwards slashes):

(4 × 3) + (−8 × 1) + (2 × 2) = 8. Then take the difference of these two numbers: |(−6 )−( 8)| = 14. Halving this gives the area of the triangle: 7. Organizing the numbers like this makes the formula easier to recall and evaluate. With all the slashes drawn, the matrix loosely resembles a shoe with the laces done up, giving rise to the algorithm's name.

• Planimeter
• Pick's theorem
• Heron's formula

• Mathologer video about Gauss' shoelace formula