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Spokane Math Circle
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The Shoelace Theorem
DG Kim
March 3rd
Introduction
The Shoelace Theorem is a nifty formula for finding the area of a polygon given the coordinates of its
vertices. In this lecture, we’ll explore the Shoelace Theorem and its applications.
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Problems without the Formula
Find the areas of the following shapes:
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Spokane Math Circle
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1.
2.
3.
4.
5.
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The Shoelace Theorem
DG Kim
March 3rd
Answers to Exercises
4 × 4 = 16
1/2 × 6 × 2 = 6
6 × 4 − 1/2 × 4 × 4 = 24 − 8 = 16
4 × 6 − 1/2 × 4 × 1 − 1/2 × 4 × 2 = 24 − 2 − 4 = 18
8 × 3 − 1/2 × 8 × 1 = 24 − 4 = 20
The Cartesian Plane
One quick note that you should know is about the Cartesian Plane. Cartesian Planes are in an (x, y)
format. The first number in a Cartesian point is the number of spaces it goes horizontally, and the
second number is the number of spaces it goes vertically. You should be able to find the Cartesian
coordinates in all the diagrams that were provided earlier.
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The Shoelace Formula!
A=
n−1
n−1
X
1 X
x
y
+
x
y
−
xi+1 yi − x1 yn i i+1
n 1
2 i=1
i=1
Okay, so this looks complicated, but now we’ll look at why the shoelace formula got its name.
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Spokane Math Circle
The Shoelace Theorem
DG Kim
March 3rd
The reason this formula is called the shoelace formula is because of the method used to find it. If
you were to find the area of a triangle with vertices (2,4), (3,-8), and (1,2), you would construct the
following matrix by ”walking around” your triangle and ending with the point you started with.
Now follow the steps taken in the diagram:
Multiply across the lines, then add up the two sides. We get 8 and -6. Now we plug them into the
formula:
1
1
× |8 − (−6)| = × 14 = 7
2
2
(The reason why this is called the shoelace formula is because of the laces pattern the cross multiplying
makes.)
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Application to our Exercises
Earlier, we found the area of all our figures by adding and subtracting areas of things we knew how to
find, like squares and triangles. Now you should find the Cartesian coordinates of all the shapes, and
apply the shoelace formula.
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Spokane Math Circle
The Shoelace Theorem
If all went well, the areas should be exactly the same as we calculated earlier.
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DG Kim
March 3rd
Spokane Math Circle
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The Shoelace Theorem
DG Kim
March 3rd
Exercises
1. A triangle has vertices at (0,0), (1,9), (4,5). What is the area?
2. Find the area in the figure:
3. In right triangle ABC, we have ∠ACB = 90◦ , AC = 2, and BC = 3. Medians AD and BE are
drawn to sides BC and AC, respectively. AD and BE intersect at point F . Find the area of 4ABF .
Prove the Pythagorean Formula using the Shoelace formula.
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Conclusion
All in all, the Shoelace theorem is a great theorem to know! It simplifies much of the work that might
have to be done slowly and meticulously. It’s a time saver, and surprisingly not that well known in the
mathematics community. Personally, I find many applications for it in many math competitions. The
Math is Cool, AMC’s, etc all can be simplied using the Shoelace theorem. 5