Rhombus and Circular Pool Write-up

by

Martha Burns

Mr. X has a rhombus shaped backyard. He wants to build the biggest possible circular pool in his backyard. How could he do it?

 Extension-1

Construct the biggest possible circular pool if the shape of the backyard is an n-sided regular polygon (square, pentagon, hexagon, etc.).

 Extension-2

Devise a plan to construct the biggest possible circular pool for any given shaped backyard.

The definition of a rhombus is a parallelogram with four equal sides consisting of two acute angles and two obtuse angles – opposite angles congruent.

Open GSP Sketchpad and draw a line segment and measure its length.

Rotate line segment from right endpoint “B” 45.0 degrees and measure its length.

Rotate line segment AB from endpoint A –135.0 degrees.  Highlight point C and construct a line parallel to line segment AB.  Mark point F at intersecting lines. Construct line segment FA and measure its length.  Construct line segment CF and measure its length.

Hide the parallel line.  Measure each angle.  Mark the interior of the rhombus and calculate its area.

Mr. X has his rhombus shaped backyard.  The measurements of the line segments and angles verify the definition of a rhombus as given in the opening statement. Construct segment AC and segment BF with red dashed lines and mark point “f” at the intersecting diagonals.

Construct Circle C1 at point “f” extending to the boundaries of the rhombus.  Label radii of circle C1 where they intersect the diagonals and measure the length of each radius.  Mark interior of circle C1 and calculate its area.

This is the biggest possible circular pool in the rhombus shaped backyard belonging to Mr. X. Calculations show the area of the pool to be less than the area of the rhombus which it should be.