Mathcounts Strategy: Shoestring (or Shoelace) method of finding the area of any polygon

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Shoelace formula from Wikipedia

More on Shoelace

Problems: Solutions below 

#1:  Find the area of a quadrilateral polygon given the four end points (3, 5), (11, 4), (7,0) and (9,8) in a Cartesian plane.

#2 2007 Chapter Target Round: A quadrilateral in the plane has vertices at (1,3),  (1,1), (2, 1) and (2006, 2007). What is the area of the quadrilateral?

#3: Find the area of a polygon with coordinates (1, 1), (3, -1),  ( 4, 4), and  (0.3)

#4: What is the number of square units in the area of the pentagon whose vertices are 

(1, 1 ), ( 3, -1),  (6, 2), (5, 6), and (2, 5)?

#5: Find the area of a polygon with coordinates ( -6, 0), (0, 5), (3, -2), and (4, 7)

#6: Find the area of a polygon with coordinates (20, 0), (0, 12), (3, 0), (4, -4)

#7: Find the area of a polygon with coordinates (-8, 4), (2, 12), (3, -5), (4, -4)

#8: Find the area of a triangle with coordinate (-8, -4), (-3, 10), (5, 6)

Solution I: Draw a rectangle and use the area of the rectangle minus the four triangles to get the area of the quadrilateral polygon. 

Solution II: Using shoestring method. First, plug in the four points. Second, choose one starting point and list the other points in order (either clockwise or counterclockwise)  and at the end, repeat the starting point. The answer is 33 square units.

Use this link to practice finding the area of any irregular polygon. Keep in mind that a lot of the times you don't need to use shoestring method. Be flexible!! Scroll to the middle section.





#2 Answer: 3008 square units

#3: Answer: 10.5 square units 

#4: Answer: 22 square units

#5: Answer: 45.5 square units

#6: Answer: 136 square units

#7: Answer: 98 square units

#8: Answer: 66 square units 

This Week's Work : Week 12 -- for Inquisitive Young Mathletes

Link to the online timed test on questions you mostly got wrong or not fast enough. 
(sent through e-mail)

Common Pythagorean Triples: 
3, 4, 5 and its derivatives 
5, 12, 13 
8, 15, 17 
7, 24, 25 (at least these for SAT I and II) 

9, 40, 41, (the rest for state and Nationals, so we'll learn them later)
11, 60, 61
12, 35, 37
13, 84, 85
20, 21, 29

Shoe string method in finding the area of any polygon

Heron's formula in finding the area of a triangle.

Don't mix up the "s" with the other "S" of finding

the area of an equilateral triangle -- proof and formula (You can also use 30-60-90 special right
triangle to get that.)
or
the area of a regular hexagon

In Heron's case, "s" stands for half of the perimeter.

Besides, I've noticed most of the questions, when given the sides, are best solved by using Pythagorean triples, especially in sprint round questions, so make sure to actively evaluate the question(s) at hand and use the most efficient strategy.

Here is the link to 2003 chapter #29 that most of you got wrong:

You don't need to use complementary counting for that specific question since it's equal cases either way. Make sure you understand why you need to times 3. (AA_, A_A, and _AA for team A to be chosen two out of three days).

From Mathcounts Mini: Area of irregular polygon

See if you can use shoestring method to get the same answer.
Second half is again on similar triangles, dimensional change and sometimes
Pythagorean triples.

From NOVA : Fractals - Hunting the Hidden Dimension

Review: Shoestring method, Heron's formula, Pythagorean Triples and Others related to Area of a triangle or polygon

Please give me feedback if there is any error or what concepts to be included for future practice tests. Thanks a lot in advance.

Online practices on finding the area of a triangle or irregular polygons.
Click here for the timed online test. Type in your first name and only write down "number" answers.

Before trying out the online timed test, here are some reminders.

a. If the irregular polygon include the origin, then using that as a starting point for shoestring method might save some calculation time.

b. There are Pythagorean Triples hidden in lots of questions that ask for the area of a triangle given three side lengths. Thus, if you find the side length that matches the triples or multiples of the triples, it might be much easier to use that instead of heron's formula.

c. For quadrilaterals, if the diagonals are perpendicular to each other, such as rhombus, kite or others, it's much easier to use D1* D2 / 2 to get the area.

d. Sometimes breaking the polygon into different smaller parts is the easiest way to find the answer.

e. Deliberate practices are the key to steady progress.

Have fun problem solving!!