Question

Prove that the ratio of the areas of any polygon circumscribed about a given circle is equal to the ratio of their perimeters.

Short answer

For any polygon circumscribed around a circle, the ratio of its area to its perimeter simplifies to \(0.5 * Radius\), regardless of its shape (number of sides and side length). Therefore, the ratio of the areas of any polygon circumscribed about a given circle is equal to the ratio of their perimeters.

Step-by-step solution

Identify the Area of a circumscribed polygon

In the case of any polygon circumscribed around a circle, the polygon can be viewed as combining multiple isosceles triangles with the circle's center as their apex. Each triangle has its base along a side of the polygon and its height as the circle's radius. The area of the overall polygon, \(A\), can then be seen as the sum of the areas of these triangles. From the general formula for the area of an isosceles triangle, \(Area = 0.5 * base * height\), we see that the total area of the polygon will be the sum of \(0.5 * Radius * Side\) for each triangle in the polygon. This can be generalized for 'n' sided polygon as \[A = 0.5 * n * Side * Radius\]

Identify the Perimeter of a circumscribed polygon

Perimeter of a polygon is simply the sum of its sides. Given that our polygon has 'n' sides of length 'Side', the total perimeter \(P\) will be \(n * Side\).

Compare the ratios of Area to Perimeter for different circumscribed polygons

Using the expressions we have derived, we see: The ratio of the area to the perimeter of the first polygon is: \[ \frac{A_1}{P_1} = \frac{0.5 * n_1 * Side_1 * Radius}{n_1 * Side_1} = 0.5 * Radius\] Similarly, for second polygon its, \[ \frac{A_2}{P_2} = \frac{0.5 * n_2 * Side_2 * Radius}{n_2 * Side_2} = 0.5 * Radius\] Notably, we see that for any polygon circumscribed around the same circle, the ratio of its area to perimeter simplifies to \(0.5 * Radius\), regardless of its details ('n' sides of 'Side' length). This establishes that the ratios of areas to perimeters for any polygon circumscribed around a given circle will indeed be equal, as intended. This proof demonstrates the mathematics behind a commonly observed geometric phenomenon and should serve to deepen understanding of the interconnection between different geometric properties.

Key concepts

Polygon

A polygon is a flat shape with straight sides. It can have any number of sides, but it must have at least three to form a closed shape. Examples of polygons include triangles, rectangles, and pentagons. Each side of a polygon meets at a vertex to form corners. The sum of the interior angles of a polygon depends on how many sides it has, calculated by the formula \((n-2) \times 180^\circ\), where \(n\) is the number of sides. Understanding polygons is essential in geometry because they form the basis for many complex shapes and problems.
Polygons can be regular or irregular. In a regular polygon, all sides and angles are equal, like in a square or equilateral triangle. In irregular polygons, sides and angles can differ.
Another important property of polygons is their symmetry, which helps in simplifying various geometric calculations.

Circumscribed Circle

A circumscribed circle, also known as a circumsircle, is a circle that passes through all the vertices of a polygon. For a polygon to have a circumscribed circle, it needs to be cyclic, which means that its vertices must all lie on a single circle.
The center of the circumscribed circle is called the circumcenter, and it can be inside or outside the polygon. The distance from the center to any vertex is the radius of the circle. For triangles, finding the circumcenter can be done by locating the intersection of the perpendicular bisectors of the sides.
Circumscribed circles are crucial in geometry as they link the properties of circles and polygons, helping solve various problems, such as those involving circumscribed polygons around circles.

Area

The area of a polygon represents the amount of two-dimensional space it occupies. Calculating the area varies depending on the type of polygon. For an isosceles triangle formed inside a circumscribed polygon, the area can be found using the formula:

• \(Area = 0.5 \times base \times height\)

Where the base is a side of the polygon and the height is the radius of the circumscribed circle.
For a polygon circumscribed around a circle, the total area can be derived by adding up the areas of all isosceles triangles formed. This gives us the formula: \(A = 0.5 \times n \times \text{Side} \times \text{Radius}\), where \(n\) is the number of sides. Understanding the area helps in determining the space covered by the polygon.

Perimeter

The perimeter of a polygon is the total length around its edges. Finding the perimeter is straightforward for any polygon, as it involves adding up the lengths of all its sides. In a polygon with regular sides, this is particularly simple and expressed as \(P = n \times \text{Side}\), where \(n\) is the number of sides.
In the context of a polygon circumscribed about a circle, it's crucial to understand that the perimeter directly influences various geometric properties, including the ratio it forms with the area. This relationship is key in the given exercise, focusing on how polygons around circles maintain specific proportional characteristics.

Isosceles Triangle

An isosceles triangle is a type of polygon with two sides of equal length, known as legs, and a third side called the base. This triangle has unique properties, such as equal base angles, making geometric calculations simpler.
In geometry, isosceles triangles are often used to solve complex problems involving relative symmetry. When considering polygons circumscribed around a circle, each side forms the base of multiple isosceles triangles within.
The common height of these triangles is the radius of the circle, which simplifies the process of calculating total area in the case of circumscribed polygons. Recognizing the isosceles triangles' role in these scenarios helps link various geometric concepts under study.