This problem was inspired from a story in my hometown newspaper about a group of neighbors who stood in a circle using social distancing guidelines during the coronavirus pandemic and shared devotionals and stories.   Click here for the story

           

If 8 people stand in a circle with six feet of required social distance between them (along a chord):
(1) What is the circumference of the circle?
(2) What is the diameter of the circle?
(3) How far in a straight line (chord) is Person A from Person C?


(4) Now, if the largest space that the neighbors have to gather in is a square 25 feet by 25 feet,
what is the maximum number of people who could stand in a circle and still be 6 feet apart?

Solution to the Problem:

(1) The circumference is 49.26 feet.
(2) The diameter is 15.68 feet.
(3) AC = 11.09 feet (7.84 times the square root of 2).
(4) 12 people could stand in the circle.

           

For problems 1 -3, construct chord BC and then draw OX perpendicular to it forming two congruent triangles.
In triangle OBX, you know that BX = 3 and the measure of angle BOX = 22.5 degrees, so you can solve for the radius R by using trigonometry.


Now consider triangle AOC.
It is an isosceles right triangle.
Since the legs (the radii) are each 7.84 feet,
then the hypotenuse AC is 7.84 times root 2 = 11.09 feet.

In problems 1 - 3, we knew the number of people and the length of a chord between 2 people, so we were able to solve for the radius and diameter.
In problem #4, we know the diameter and the chord, so we are looking for the degree measure of an arc (which equals the central angle) and then we can divide that into 360 degrees to see how many people can be in the circle and still be 6 feet apart.

We can use triangle BOX above but this time we know R = 12.5 feet and BX = 3 feet and we will solve for the angle BOX.
sin (BOX) = 3 / 12.5, so angle BOX = 13.89 degrees.
So, angle BOC = 27.77 degrees which is the degree measure between two adjacent people.
Now divide 360 degrees by 27.77 degrees to get 12.96 people.
If you are using the strict guidelines of 6 feet apart, then 12 people is the answer.
But maybe you could squeeze 13 people into the circle.


Correctly solved by:

1. Colin (Yowie) Bowey Beechworth, Victoria, Australia
2. John Simmons Memphis, Tennessee
3. Ivy Joseph Pune, Maharashtra, India