

Circumscribed rectangle, or bounding box(Coordinate Geometry)
The smallest
rectangle that contains all of the given points.
Try this
Drag any of the four points below. The bounding box will adjust accordingly.
The circumscribed rectangle, or bounding box, is the smallest rectangle that can be drawn around a set of points such
that all the points are inside it, or exactly on one of its sides.
The four sides of the rectangle are always either vertical or horizontal, parallel to the x or y axis.
In the figure above, the bounding box is shown drawn around the vertices of a quadrilateral ABCD.
It is called the bounding box because it forms a boundary, like a fence, around the shape or set of points.
This is used extensively when finding areas of various shapes using coordinate geometry.
(For example see Area of a triangle (box method) )
This method involves first drawing the bounding box,
and then subtracting the areas of simple shapes created around the edge of it to find the area of the desired figure.
Click on 'reset' in the figure above and note that:
 The top of the rectangle is determined by the ycoordinate of the topmost point  point B
 The bottom of the rectangle is determined by the ycoordinate of the lowest point  point D
 The left side of the rectangle is determined by the xcoordinate of the leftmost point  point A
 The right of the rectangle is determined by the xcoordinate of the rightmost point  point C
Area and perimeter
By finding the coordinates of the corners of the box, you can find its width, height, area and perimeter.
See
Example
Find the area of the circumscribed rectangle in the figure above.
Click on "reset" and "show coordinates".

As you can see, the top of the rectangle is determined by the point nearest the top,
the one with the largest ycoordinate. In this case it is point B with a ycoordinate of 40.
All points along the top of the rectangle therefore have a ycoordinate of 40.

The left side of the rectangle is defined by point A, the one with the lowest xcoordinate.
All points down the left side of the rectangle therefore have an xcoordinate of 10.

The top left corner must therefore have an xcoordinate of 10 and y coordinate of 40. Or (10,40).
We repeat this process for the other 3 corners to get the result below. Click on "show pointers' in the figure to help visualize this.
Top Left 
(10,40) 
Top Right 
(60,40) 
Bottom Left 
(10,11) 
Bottom Right 
(60,11) 
 The height of the box is the difference between the y coordinates of any top and bottom point. Here, that is 4011 or 29.
 The width is the difference between the xcoordinates of any left or right points.
Here, that is 6010, or 50.
 The area of the box is the width times height. Here, 29 times 50, or 1450.
 The perimeter of the box is twice the width plus height. Here, that is 2(29+50), or 158.
Things to try

In the above diagram, press 'reset'.

Drag the point A to the right so it is inside the box and no longer determines the box size.

Drag any of the points A,B,C,D around and note how the points control the bounding box.
Calculate the coordinates of the four corners. Click on "show pointers" to verify the result.
Limitations
In the interest of clarity in the applet above, the coordinates are rounded off to integers and the lengths rounded to one decimal place.
This can cause calculatioons to be slightly off.
For more see
Teaching Notes
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