Can you fit more people around a round table or a square table with the same surface area?

Answer 1

If we want to look at the relationship of a square and a circle, it helps to use an example.

Imagine we have a circle table and square table both with areas of 100.

The square has equal sides, thus the area is achieved by squaring one of these sides. This means the side of the square is the square root of the area which is 10. If we multiple this side by 4 we get a perimeter of 40.

For the circle it is a bit trickier since we have an irrational number to deal with, namely pi (π). We will need to do a bit of algebra to arrive at the measurements of this circle from an area of 100.

The area of a circle is calculated using the formula π r ² . If we know the area equals 100, we can turn this formula into an equation π r ² = 100 which then becomes r ² = 100/π which then becomes r = √(100/π).

Now, that we have a "value" for the radius of this circle we can plug it into the formula to determine the circumference (perimeter) of the circle. C = 2 π r Now, we substitute our new value of r in there and we get: C = 2 π √(100/π). If we substitute 3.1416 for π, we get an approximate circumference of 35.449.

This example shows that a square table provides more perimeter for seating than a circle table of equal area.

In fact, the ratio of the perimeters remains constant regardless of how large or small you grow the area of the 2 tables. The circumference (perimeter) of the round table will always be about 88.6% of the perimeter of the square table of equal area.

Answer 2

More people can fit around a square table than a round table of the same area. Since the circumference of any circle is C = (3.14)(diameter). Lets take a circle with a diameter of 10 feet; it's circumference then is 31.4 ft., and its area is A = (3.14)(5)² = 78.5 ft².

A square table having the same area would have each side equal to √78.5 ft² = 8.86 ft. Let's say that two people can fit on each side of the square table; then the table can seat 8 people, allowing each person to occupy 4.41 ft. When we ask how many people, each requiring 4.41 ft, can fit around the circle with a circumference of 31.4 ft, we find 31.4 divided by 4.41 = 7.12 people. Therefor, more people fit around a square table than a cicular table with the same area.