Exploring the Ratio of the Areas of a Square and a Regular Hexagon Inscribed in a Circle

When it comes to understanding the geometric properties of shapes inscribed in circles, the ratio of their areas is an intriguing area of study. This article will delve into the areas of a square and a regular hexagon inscribed within a circle, providing a detailed explanation of the calculations involved.

Introduction to Inscribed Geometric Shapes

Geometric shapes inscribed in a circle have a unique relationship with the circle's properties. The area of a shape inscribed in a circle is directly related to the circle's radius. In this article, we will focus on the areas of a square and a regular hexagon inscribed in the same circle and explore the ratios between these areas.

Area of the Square Inscribed in a Circle

Consider a square inscribed within a circle of radius (r). To find the area of the square, we start by understanding the relationship between the square's diagonal and the circle's diameter.

Area of the Square

The diagonal of the square, which is also the diameter of the circle, is given by:

[d 2r]

The side length (s) of the square can be related to the diagonal through the Pythagorean theorem:

[ssqrt{2} d]

Substituting (d 2r):

[ssqrt{2} 2r]

Therefore:

[s frac{2r}{sqrt{2}} rsqrt{2}]

The area (A_s) of the square is:

[A_s s^2 (rsqrt{2})^2 2r^2]

Area of the Regular Hexagon Inscribed in a Circle

In a regular hexagon inscribed in a circle, each side length (a) is equal to the radius (r) of the circle. The area (A_h) of a regular hexagon inscribed in a circle can be calculated using the formula:

[A_h frac{3sqrt{3}}{2} a^2]

Substituting (a r)

Substituting (a r) into the formula:

[A_h frac{3sqrt{3}}{2} r^2]

Ratio of the Areas

Now, let us find the ratio of the area of the square to the area of the hexagon.

[text{Ratio} frac{A_s}{A_h} frac{2r^2}{frac{3sqrt{3}}{2} r^2}]

The (r^2) terms cancel out:

[text{Ratio} frac{2}{frac{3sqrt{3}}{2}} frac{2 cdot 2}{3sqrt{3}} frac{4}{3sqrt{3}}]

Simplifying further, we multiply the numerator and denominator by (sqrt{3}):

[text{Ratio} frac{4sqrt{3}}{9}]

Conclusion

The ratio of the areas of a square to a regular hexagon both inscribed in a circle is:

[frac{4sqrt{3}}{9}]

Additional Observations

For a square with each side of length (rsqrt2), the area is (2r^2).

A regular hexagon can be divided into 6 equilateral triangles, each with an area of (frac{r^2}{2sqrt{3}}), resulting in a total area of:

[6 times frac{r^2}{2sqrt{3}} frac{3sqrt{3}}{2} r^2]

The ratio of the areas of the square to the hexagon is also:

[frac{2r^2}{frac{3sqrt{3}}{2} r^2} frac{4}{3sqrt{3}}]

This confirms our previous calculation, providing a deeper understanding of the geometric properties and relationships involved.