Sukkah Size It!

Sukkot_in_Jerusalem,_1912 short

In parsha Emor (Vayikra 23:42) we are commanded to sit in a succah on Succos.

Author: Shlomo Miller
Math Level: Geometry
Grade Level: 8th
Topics: Relationship between circumference and area of a circle


The Question:
How do you find the circumference of a sukkah that has the same area as the minimum-sized square sukkah?

The Answer:
Tosfos uses a mashal. Imagine you have a circle of a certain size. Fill that circle with concentric pieces of string, with the largest string going around the circumference of the circle and the smallest going right near the center.

Then imagine that you cut the strings along one radius of the circle and stretch them out, forming a triangle. The largest string, with a length equivalent to the circumference of the circle, will be the base of the triangle. The string next to it will be slightly smaller than it, and the one next to it even smaller until the point of the triangle is reached.

The height of the resultant triangle is the radius of the circle, and its base is the circumference. The area of the triangle is now .5*radius*circumference. Since the circle and triangle have equivalent areas, the area of the circle is also .5*radius*circumference.

In a previous post we determined that the circumference of a circle is 3*diameter, or 3*2*radius. Substituting that in, the area of a circle is 3*radius^2.