Open Middle®
Directions: Which quadrilateral has a greater area? Quadrilateral A has its perimeter equal to 44 units. Quadrilateral B has the sum of its interior angles equal to 360 degrees.
Hint
What is the greatest possible area for a quadrilateral with a perimeter of 44 units? What is the greatest possible area for a quadrilateral whose interior angles have a sum of 360 degrees?
Answer
This may appear to be a trick question, but if students have conceptual understanding of quadrilaterals, they should know that all quadrilaterals have the sum of their interior angles equal to 360 degrees so we don’t have enough information to determine which quadrilateral has a greater area.
Source: Robert Kaplinsky
Since angles tell you about shape and perimeter tells you about size it would be perhaps a more exciting problem if you gave then 2 or even 3 angle measures for quad B
Thanks Howard. Can you tell me more about what you mean? I’ve having trouble envisioning what that would look like.
Or perhaps, ask what is the greatest possible area for a quadrilateral with a perimeter of 44 units?
Thanks Ryan. We already have a problem like that here: https://www.openmiddle.com/rectangles-maximizing-area/
If the question were changed the next day to read which quadrilateral has the potential to have the biggest area, we might see some interesting conceptual discussions.
Since there are no specific parameters, “What is the greatest possible area for a quadrilateral whose interior angles have a sum of 360 degrees?” For a rectangle, one side could be infinity minus 1, the other side could be two, therefore the area is infinite!
The 360 degrees at first confused me but then I got it. Nice tough question
Yeah it was a little confusing at first but then I got it. It was pretty cool!
Nice question. This is a trick question, though. We don’t know enough about either quadrilateral. All quadrilaterals have the sum of their interior angles to be 360, which doesn’t tell us anything about their area. There can be multiple areas for Quad A. An easy way to remember the sum of a polygon is by dividing the shape into the least amount of triangles and adding 180 for each triangle. Ex: Triangle:180, Quadrilateral:360, Pentagon:540, etc.