Understanding Inscribed Polygons: A Mathematical Puzzle

In this article, we explore a fascinating problem that combines geometry and trigonometry. Given a circle, we inscribe both a square and a hexagon within it, and we uncover a surprising relationship between their apothems. This exploration not only enriches our understanding of geometric properties but also provides practical insights into the behavior of regular polygons inscribed in circles.

The Problem: Inscribed Square and Hexagon with Different Apothems

We are tasked with a problem: what is the radius of the circle in which a square and a hexagon are inscribed, given that their apothems differ by 1? This requires us to delve into the geometric properties and trigonometric formulas of these regular polygons.

Definitions: Apothem and Radius

Before we proceed, let's define the terms we will use: Apothem: The distance from the center to the midpoint of one of the sides of a regular polygon. Radius: The distance from the center of the polygon to any of its vertices.

Formulas for Apothem: Square and Hexagon

To solve the problem, we need to use the formulas for the apothem of a square and a regular hexagon.

The apothem of a square with side length ( s ) is given by:

( a_s frac{s}{2} cdot frac{1}{tanfrac{pi}{4}} frac{s}{2} )

The side length ( s ) of the square is related to the radius ( R ) of the circumscribed circle:

( s R sqrt{2} )

Therefore, the apothem of the square is:

( a_s frac{R sqrt{2}}{2} )

The apothem of a regular hexagon with side length ( s ) is given by:

( a_h frac{s}{2} cdot frac{1}{tanfrac{pi}{6}} frac{s sqrt{3}}{3} )

The side length ( s ) of the hexagon is equal to the radius ( R ) of the circumscribed circle:

( s R )

Therefore, the apothem of the hexagon is:

( a_h frac{R sqrt{3}}{2} )

Given Condition: Apothems Differ by 1

Given that the apothems differ by 1, we have:

( a_s - a_h 1 )

Substituting the expressions for the apothems, we get:

( frac{R sqrt{2}}{2} - frac{R sqrt{3}}{2} 1 )

This simplifies to:

( frac{R}{2} (sqrt{2} - sqrt{3}) 1 )

Solving for ( R ), we get:

( R (sqrt{2} - sqrt{3}) 2 )

( R frac{2}{sqrt{2} - sqrt{3}} )

Final Calculation

Calculating ( sqrt{2} ) and ( sqrt{3} ): ( sqrt{2} approx 1.414 ) ( sqrt{3} approx 1.732 )

( sqrt{2} - sqrt{3} 1.732 - 1.414 0.318 )

Substituting back into the formula for ( R ), we get:

( R frac{2}{0.318} approx 6.283 )

Conclusion

The radius of the circle is approximately 6.283, which means the circle can accommodate both a square and a hexagon with their apothems differing by 1 unit. This solution not only provides a concrete answer to a geometric problem but also illustrates the interplay between trigonometric functions and the properties of regular polygons. The combination of these concepts showcases the elegance and interconnectedness of mathematics.

Practical Insight

By understanding the relationship between the apothems of inscribed polygons and the radius of the circumscribed circle, we can gain insights into the geometric properties of regular polygons. This knowledge can be applied in various fields, including architecture, design, and engineering, where precise geometric relationships are crucial. In conclusion, this problem illustrates how mathematical principles can provide elegant solutions to seemingly complex problems. By breaking down the problem into clear, manageable parts, we can uncover deeper mathematical truths.