How do you find the perimeter of a square on a grid and the vertical of (1,1) (1,6) (4,6) (4,1)

Question

I dont knowhow to find the perimeter of a polygon with vertics

Answer 1

Finding the perimeter of any polygon is a matter of adding the length of the line segments that form its sides.  So, if a four-sided figure has vertices A, B, C, and D, then the perimeter is the sum of the lengths AB, BC, CD, and DA:

           Perimeter = AB + BC + CD + DA

That would be O.K. for a square (for which AB=BC=CD=DA) or a rectangle (for which AB=CD and BC=DA) or any other four-sided figure.

 

And -- the same process works for a five-sided figure or a six-sided figure or a ...  (well, you get the idea).  Often, it is just a matter of patience to calculate the perimeter.  This is important, for example, when you want to put a fence around some property that has a complicated shape -- it is unacceptable to come up short of materials or to have way too much fence left over; and you would like to estimate the time it will take so you can get an idea of the total cost of the project (or share half the cost with each neighbor).

 

On a grid, there is a very important, very interesting help -- the distance formula.  Two points, say A and B, have an x-distance difference and a y-distance difference that forms a right triangle with the hypotenuse being the direct distance.  So, using the Pythagorean Theorem, the distance formula gives us the direct distance between any two points:

                      Direct distance = SQRT(  (x-difference)^2  +  (y-difference)^2 )

    For the points A(1,1) and B(1,6)  this is:

                      Distance AB = SQRT( (1-1)^2 + (6-1)^2 )

                                         = SQRT(   0   + 5^2)

                                         =  5

 

Note:  When two points have the same x-coordinate, they form a vertical line segment.  When two points have the same y-coordinate, they form a horizontal line segment.  Those distances are pretty easy to determine.  It's the slanted lines that force us to use the distance formula.

Answer 2

Sam- I would graph the points.  You will be able to see the distance between the corners of the shape and determine the overall perimeter.  The numbers you provided should actually make a rectangle (rather than a square)

 

Alternatively, you can think about the relationship between two points.

• (1,1) and (1,6) share the same x-coordinate (1).  They y coordinates are 5 units apart (6-1).  This means the left side of the shape has a length of 5 units.
• from (1,6) to (4,6).  These share the same y-coordinate (6).  The x-coordinates are 3 units apart (4-1).  The top of the shape has a length of 3 units.
• From (4,6) to (4,1).  These points share an x-coordinate (4), so you are looking at the difference between the y-coordinates (6-1).  So the length of the right side of the shape is 5 units.
• From (4,1) to (1,1).  This will close our quadrilateral shape.  The y-coordinates are the same (1), so we are looking at the difference in the x coordinates (4-1).  The length of the bottom of the shape is 3 units.

 

• The perimeter of the shape is 5+3+5+3.  The total perimeter is 16 units, and since there are two sets of equal sides at right angles, this is a rectangle.

Answer 3

Unfortunately, this is not a square, but it is a rectangle so still reasonable to solve. We look not at the vertices, but the distance between the vertices when trying to find lengths. The distance between (1,1) and (1,6) is 5. The distance between (1,6) and (4,6) is 3. The distance between (4,6) and (4,1) is 5, and the distance between (4,1) and (1,1) is 3. Thus the perimeter is 5+3+5+3 = 16. If uncertain as to how the lengths were determined, draw the points on a grid and connect the dots to make a rectangle and see the length of the sides.