problem solving with trigonometry **Problem 6: Calculating the Area within a Circle but outside an Inscribed Regular Hexagon**Given a regular hexagon with side lengths of 16 cm, inscribed in a circle, the task is to determine the area enclosed by the circle but lying outside the hexagon. In order to solve this problem, follow these steps:1. **Calculate the Radius of the Circumscribed Circle:** - For a regular hexagon inscribed in a circle, the radius of the circle is equal to the length of the side of the hexagon. Therefore, the radius \(r\) of the circle is 16 cm.2. **Find the Area of the Circle:** - The formula for the area of a circle is \(A = \pi r^2\). - Substitute the value of \(r\) to get the area: \[ A = \pi \times (16)^2 = 256\pi \text{ cm}^2 \]3. **Calculate the Area of the Regular Hexagon:** - The formula for the area of a regular hexagon with side length \(s\) is \(\frac{3\sqrt{3}}{2} s^2\). - Substitute the value of \(s\) to find the area: \[ A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} \times (16)^2 = 384\sqrt{3} \text{ cm}^2 \]4. **Determine the Area between the Circle and the Hexagon:** - Subtract the area of the hexagon from the area of the circle: \[ A_{\text{region}} = 256\pi - 384\sqrt{3} \text{ cm}^2 \]This final value represents the area inside the circle but outside the hexagon.

Given : Sides of a regular hexagon = 16cm

Answered: 6. A regular hexagon whose sides are 16…

6. A regular hexagon whose sides are 16 cm is inscribed in a circle. Find the area inside the circle and outside the hexagon.

Trigonometry (11th Edition)

11th Edition

ISBN:9780134217437

Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Chapter1: Trigonometric Functions

Section: Chapter Questions

Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.

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Question

problem solving with trigonometry

**Problem 6: Calculating the Area within a Circle but outside an Inscribed Regular Hexagon**

Given a regular hexagon with side lengths of 16 cm, inscribed in a circle, the task is to determine the area enclosed by the circle but lying outside the hexagon. In order to solve this problem, follow these steps:

1. **Calculate the Radius of the Circumscribed Circle:**
- For a regular hexagon inscribed in a circle, the radius of the circle is equal to the length of the side of the hexagon. Therefore, the radius \(r\) of the circle is 16 cm.

2. **Find the Area of the Circle:**
- The formula for the area of a circle is \(A = \pi r^2\).
- Substitute the value of \(r\) to get the area:
\[
A = \pi \times (16)^2 = 256\pi \text{ cm}^2
\]

3. **Calculate the Area of the Regular Hexagon:**
- The formula for the area of a regular hexagon with side length \(s\) is \(\frac{3\sqrt{3}}{2} s^2\).
- Substitute the value of \(s\) to find the area:
\[
A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} \times (16)^2 = 384\sqrt{3} \text{ cm}^2
\]

4. **Determine the Area between the Circle and the Hexagon:**
- Subtract the area of the hexagon from the area of the circle:
\[
A_{\text{region}} = 256\pi - 384\sqrt{3} \text{ cm}^2
\]

This final value represents the area inside the circle but outside the hexagon.

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