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 sides of the rectangle are always parallel to the axes. In the figure above, the bounding box is shown as the blue shaded rectangle.

This is used extensively when finding areas of various shapes using coordinate geometry. (See the area-related pages in this section)

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

Area and perimeter

• Area of a rectangle (Coordinate Geometry)
• Perimeter of a rectangle (Coordinate Geometry)

Example

Find the coordinates of the four corners 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 D with a y-coordinate of 30. All points along the top of the rectangle therefore have a y-coordinate of 30.
2. The left side of the rectangle is defined by point C, the one with the lowest x-coordinate. All points along the left side of the rectangle therefore have an x-coordinate of 7.
3. The top left corner must therefore have an x-coordinate of 7 and y coordinate of 30. Or (7,30).

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	(7,30)
Top Right	(44,30)
Bottom Left	(7,8)
Bottom Right	(44,8)

Things to try

1. In the above diagram, press 'reset'.
2. Drag the point A in such a way as to leave another point inside the rectangle.
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.

Other Coordinate Geometry entries

• Introduction to coordinate geometry
• The coordinate plane
• Coordinates of a point
• Distance between two points

• Introduction to Lines in Coordinate Geometry
• Lines (in Coordinate Geometry)
• Rays (in Coordinate Geometry)
• Line Segments (in Coordinate Geometry)
• Midpoint of a line segment (Midpoint Theorem)
• Slope (m) of a line
• Intercept (b) of a line
• Equation of a line in slope-intercept form y=mx+b
• Equation of a line in point-slope form y=m(x-Px)+Py

• Vertical lines
• Horizontal lines
• Parallel lines
• Perpendicular lines
• Intersecting lines

• Area of a triangle (formula method)
• Area of a triangle (box method)

• Print blank graph paper