Circumscribed rectangle, or bounding box (Coordinate Geometry)

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 y-coordinate of the top-most point - point B
• The bottom of the rectangle is determined by the y-coordinate of the lowest point - point D
• The left side of the rectangle is determined by the x-coordinate of the leftmost point - point A
• The right of the rectangle is determined by the x-coordinate of the rightmost point - point C

Area and perimeter

• Area and Perimeter of a rectangle (Coordinate Geometry)

Example

Find the area of the circumscribed rectangle in the figure above. Click on "reset" and "show coordinates".

1. As you can see, the top of the rectangle is determined by the point nearest the top, the one with the largest y-coordinate. In this case it is point B with a y-coordinate of 40. All points along the top of the rectangle therefore have a y-coordinate of 40.
2. The left side of the rectangle is defined by point A, the one with the lowest x-coordinate. All points down the left side of the rectangle therefore have an x-coordinate of 10.
3. The top left corner must therefore have an x-coordinate 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.

• The height of the box is the difference between the y coordinates of any top and bottom point. Here, that is 40-11 or 29.
• The width is the difference between the x-coordinates of any left or right points.
  Here, that is 60-10, 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

1. In the above diagram, press 'reset'.
2. Drag the point A to the right so it is inside the box and no longer determines the box size.
3. 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

Other Coordinate Geometry topics

• Introduction to coordinate geometry
• The coordinate plane
• The origin of the plane
• Axis definition
• Coordinates of a point
• Distance between two points

• Introduction to Lines
  in Coordinate Geometry
• Line (Coordinate Geometry)
• Ray (Coordinate Geometry)
• Segment (Coordinate Geometry)
• Midpoint Theorem

• Distance from a point to a line
• - When line is horizontal or vertical
• - Using two line equations
• - Using trigonometry
• - Using a formula

• Intersecting lines

• Cirumscribed rectangle (bounding box)
• Area of a triangle (formula method)
• Area of a triangle (box method)
• Centroid of a triangle
• Incenter of a triangle
• Area of a polygon
• Algorithm to find the area of a polygon
• Area of a polygon (calculator)
• Rectangle
• Definition and properties, diagonals
• Area and perimeter
• Square
• Definition and properties, diagonals
• Area and perimeter
• Trapezoid
• Definition and properties, altitude, median
• Area and perimeter
• Parallelogram
• Definition and properties, altitude, diagonals

• Print blank graph paper