Geometry, Baking & Decorating Cakes…

A colleagues began a new venture last summer…Rich in Blessings, baking cakes.  She sent a link to this post
Perfect Buttercream Stripes to share the math needed to complete the task shown below.

Here are a few images in the post:

There could be some simple, yet nice questions arise in this setting, like:

If Mrs.D wanted 1.5″ stripes on a 10″ cake, approximately what cental angle measure would she need to use? 

If she chose the slanted stripes with 1″ width, what angles would result in the guide strip for an 8″ cake?

If she used a 30º central angle for a 12″ cake, how wide would she need to cut the guiding stripes?

If a batch of buttercream covers ____ 10″ smooth cakes, how many batches would I need to decorate ____ 12″ cakes.

Not sure at what level this is in the standards but I plan to sit down this afternoon and determine how I can use this context during the spring semester…I believe I can make this work for C.A.1, C.A.2 and C.B.5…

From corestandards.org

Understand and apply theorems about circles

CCSS.Math.Content.HSG-C.A.1 Prove that all circles are similar. 

This could easily be done by constructing a “cake map” including 6″, 8″, 10″, 12″.  Allowing students to prove that all of their circles are similar, by showing they are dilations of one another.  Maybe I could even ask, if I wanted to enlarge my diameter 2″, what scale factor would be needed to accomplish this?

CCSS.Math.Content.HSG-C.A.2 Identify and describe relationships among inscribed angles, radii, and chords. 
Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. (Thoughts here?)

CCSS.Math.Content.HSG-C.A.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

CCSS.Math.Content.HSG-C.A.4 (+) Construct a tangent line from a point outside a given circle to the circle.

Find arc lengths and areas of sectors of circles

CCSS.Math.Content.HSG-C.B.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

Maybe play off the idea: use different colors of fondant to create a pattern of the sectors…how many units^2 of each color are needed?

I believe with a bit of work, this could qualify as my Practical Living and Career Studies Program Review submission.  If I plan efficiently, collaborating with my Visual Arts department, I may be able to use it as the Arts/Humanities submission as well.

What ideas, suggestions can you offer that will push my thinking forward…make this a good, quality task?