Question

Find the ratio of the area of the regular polygon of 12 sides circumscribed about a circle to the area of the regular polygon of the same number of sides inscribed in the circle.

Short answer

Answer: The ratio of the area of the regular polygon of 12 sides circumscribed about a circle to the area of the regular polygon of the same number of sides inscribed in the circle is \(\frac{1}{\cos^2(15)}\).

Step-by-step solution

Calculate the side lengths of the inscribed and circumscribed polygons

Using the formulas mentioned in the analysis, we calculate the side lengths of the inscribed and circumscribed polygons with 12 sides based on the radius of the circle. Let r be the radius of the circle for the inscribed polygon and R for the circumscribed polygon. We have: Inscribed polygon side length: \(s_{in} = 2r \sin(180/12)\) Circumscribed polygon side length: \(s_{cir} = 2R \sin(180/12)\)

Calculate the areas of the inscribed and circumscribed polygons

With the side lengths calculated, we can find the areas of both regular polygons using the area formula: Inscribed polygon area: \(A_{in} = \frac{12 \cdot s_{in}^2}{4 \cdot \tan(180/12)}\) Circumscribed polygon area: \(A_{cir} = \frac{12 \cdot s_{cir}^2}{4 \cdot \tan(180/12)}\)

Find the ratio of the areas

To find the ratio of the areas of the circumscribed polygon to the inscribed polygon, we divide \(A_{cir}\) by \(A_{in}\): \(Ratio = \frac{A_{cir}}{A_{in}}\) Simplify the ratio using the expressions for \(A_{cir}\) and \(A_{in}\): \(Ratio = \frac{\frac{12 \cdot s_{cir}^2}{4 \cdot \tan(180/12)}}{\frac{12 \cdot s_{in}^2}{4 \cdot \tan(180/12)}}\) The terms with 12 and \(\tan(180/12)\) cancel out: \(Ratio = \frac{s_{cir}^2}{s_{in}^2}\) Since \(s_{cir} = 2R \sin(180/12)\) and \(s_{in} = 2r \sin(180/12)\): \(Ratio = \frac{(2R \sin(180/12))^2}{(2r \sin(180/12))^2}\) The terms with \(2\sin(180/12)\) cancel out: \(Ratio = \frac{R^2}{r^2}\) Now, using the formula for circumscribed and inscribed polygons, \(R = \frac{r}{\cos(\frac{180}{12})}\), or \(R^2 = \frac{r^2}{\cos^2(\frac{180}{12})}\) We substitute this value for \(R^2\) in the previous ratio expression: \(Ratio = \frac{\frac{r^2}{\cos^2(\frac{180}{12})}}{r^2}\) The term \(r^2\) cancels out: \(Ratio = \frac{1}{\cos^2(\frac{180}{12})}\)

Calculate the final value of the ratio

Finally, we can substitute the angle value into the cosine expression to find the ratio of the areas: \(Ratio = \frac{1}{\cos^2(15)}\) Hence, the ratio of the area of the regular polygon of 12 sides circumscribed about a circle to the area of the regular polygon of the same number of sides inscribed in the circle is \(\frac{1}{\cos^2(15)}\).

Key concepts

Inscribed and Circumscribed Polygons

Imagine drawing a regular polygon inside a circle, where each vertex of the polygon touches the circle. This is known as an **inscribed polygon**. These polygons hug the circle closely, with all their vertices lying on the circumference. When the circle is fully inside the polygon with each side tangent to the circle, it's a **circumscribed polygon**. This means the polygon encloses the circle.

The main distinction between these two is how they wrap around the circle:

• **Inscribed polygons** have their vertices touching the circle.
• **Circumscribed polygons** have each side tangential to the circle.

These geometric configurations are essential when calculating areas and understanding relationships between different shapes in geometry.

Area of Polygons

Calculating the area of a regular polygon involves understanding its side lengths and angles. In our problem, both the inscribed and circumscribed polygons have 12 sides, but they differ in how they interact with the circle.

The area of a regular polygon can be calculated using its side length. Here's how:

• **Inscribed Polygon Area**: For a polygon inscribed in a circle, the area formula involves its side length and trigonometric functions relating to its angles.
• **Circumscribed Polygon Area**: This uses a similar formula, but the side length comes from the polygon's external interaction with the circle.

To find the area of either polygon, use the formula: \[ A = \frac{n \cdot s^2}{4 \cdot \tan(\frac{180}{n})} \] where \( n \) is the number of sides, and \( s \) is the side length. This formula incorporates the polygon's side length with its angular components through trigonometric functions.

Trigonometric Ratios

To deeply understand polygons and circles, we rely on trigonometric functions like sine, cosine, and tangent. These functions relate the angles of the polygons to their side lengths and eventually help in area calculations.

Consider the following aspects:

• **Sine and Cosine in Polygons**: They help determine the lengths of the sides of inscribed and circumscribed polygons. Specifically, \( \sin \left( \frac{180}{n} \right) \) is used to calculate side lengths of polygons derived from angles within the circle.
• **The role of \( \cos \)**: We used the identity relating \( R \) and \( r \) as \( R = \frac{r}{\cos(\frac{180}{n})} \). It demonstrates how \'cosine\' provides the link between the radius of circumscribed and inscribed polygons.

Understanding these trigonometric relationships is crucial for solving geometry problems involving regular polygons and circles.