When Area equals Perimeter (in numerical value).

Here's a cute little topic: When does a two-dimensional geometric figure have area and perimeter the same numerical value? There are classic examples of integer rectangles and integer right triangles with this property. An email from Jim M. inspired me to push this all a little bit further. (Thank you, Jim!)

[willjohnston]

So much content in such a short amount of time. Reviews area of triangle, perimeter of regular figures using apothem, use of tangent ratio in regular polygons, and even an informal limit approach showing that circle follows the pattern of regular polygons. I like the interspersed thinking challenges put in at natural pause points. Something for students at many levels, gives everyone an opportunity to take something away.

[magicjim]

Outstanding content as always!

After all of the work I have done on this, I never noticed the radius of 2 being the radius for the corresponding incircle (if there is one).

I also played around with figuring out the radius of a sphere inscribed in a polyhedron. The radius must equal the volume of the polyhedron divided by one-third of the polyhedron's surface area.

[srikanthdrao]

For any dimension 'n' the 'hyper'sphere with radius 'n' units will have its 'surface area' and 'volume' equal 'numerically'.
Also the inscribed hypersphere inside the hypercube will have it's surface area and volume equal to eachother numerically.
So for 2D, the radius is 2 units and for 3D, the radius was 3 units and so on.

[benhbr]

10:45 that converse (A = P ⇒ r = 2) is yet to be proved