I gave this question to students as a challenge at the end of a trig unit.

A quadrilateral has perimeter = 60 and a 30º angle. What is the maximum possible area?

I think this is cute. The kids had to make some assumptions, test them, and use trigonometry along the way. It’s not “open-ended” but it does involve some investigation, and it is not just a direct application of what I’ve taught them.

Do you like the question?

And do you know why “What is the minimum possible area?” is not a good question?

1. April 7, 2018 am30 11:52 am 11:52 am

Had to make a GeoGebra sketch for looking into this. Trying to figure out how to pitch it for my calculus students. https://www.geogebra.org/m/ZDJbE4tK

• April 7, 2018 pm30 6:43 pm 6:43 pm

I did this with students without calculus, we made arguments that appealed to the maximum value of sine, maximum altitudes, and to symmetry.

• April 10, 2018 am30 9:46 am 9:46 am

As a calculus problem it’s very challenging! Although I also feel like the symmetry arguments are tricky, because of that fixed angle. (The kind of sliding around of things I want to do tends to disrupt angles.)

May 1, 2018 pm31 2:19 pm 2:19 pm

Try setting AB to 20, and AD to 0.001.

April 12, 2018 am30 8:14 am 8:14 am

It works with a rhombus, which is a skewed square. A square maximizes area. Therefore each side should be equal to 15 units where the angles are 30, 30, 150 and 150. Using the trig formula of A = (s)(s) sine (angle), you get a maximum area of 112.5 square units. You cannot minimize area because of the constraints of the angle and perimeter. Whadda think?

3. May 1, 2018 pm31 4:51 pm 4:51 pm

I posted this question on MathStackExchange [https://math.stackexchange.com/q/2760727] where it was pointed out that there should be an assumption around convexity: otherwise, you can create a concave figure that approaches an equilateral triangle to maximize the area. See the link above and its answers – some of which contain some tough mathematics! – for more information [including some generalizations].

• June 12, 2018 pm30 9:30 pm 9:30 pm

Thank you for extending the discussion, and exhausting the issues with the problem statement. The stackexchange discussion was interesting for me.