7) regular dodecagon 9) regular 16 -gon regular 24-gon 10) regular 13-gon

Certainly! Let's delve into the properties of the specified regular polygons: --- ### 7) Regular Dodecagon (12-gon) **Definition:** A dodecagon is a 12-sided polygon. A **regular dodecagon** has all sides of equal length and all interior angles equal. **Properties:** - **Interior Angle:** \[ \text{Each interior angle} = \frac{(n - 2) \times 180^\circ}{n} = \frac{(12 - 2) \times 180^\circ}{12} = 150^\circ \] - **Exterior Angle:** \[ \text{Each exterior angle} = 180^\circ - 150^\circ = 30^\circ \] Alternatively, since the sum of exterior angles is \(360^\circ\): \[ \frac{360^\circ}{12} = 30^\circ \] - **Sum of Interior Angles:** \[ (n - 2) \times 180^\circ = 10 \times 180^\circ = 1800^\circ \] - **Area:** Given side length \(s\), the area \(A\) can be calculated using: \[ A = \frac{3n \times s^2}{4 \tan\left(\frac{\pi}{n}\right)} = 3 \times \frac{s^2}{4 \tan\left(\frac{\pi}{12}\right)} \approx 3 \times \frac{s^2}{4 \times 0.2679} \approx 3 \times \frac{s^2}{1.0716} \approx 2.8s^2 \] *(Note: For precise calculations, use exact trigonometric values or a calculator)* - **Symmetry:** - **Rotational Symmetry:** 12-fold. - **Reflection Symmetry:** 12 axes of symmetry. - **Circumradius (\(R\)):** The radius of the circumscribed circle: \[ R = \frac{s}{2 \sin\left(\frac{\pi}{12}\right)} \approx \frac{s}{2 \times 0.2588} \approx \frac{s}{0.5176} \] --- ### 9) Regular 16-gon and Regular 24-gon #### **Regular 16-gon (Hexadecagon)** **Properties:** - **Interior Angle:** \[ \frac{(16 - 2) \times 180^\circ}{16} = \frac{14 \times 180^\circ}{16} = 157.5^\circ \] - **Exterior Angle:** \[ \frac{360^\circ}{16} = 22.5^\circ \] - **Sum of Interior Angles:** \[ (16 - 2) \times 180^\circ = 14 \times 180^\circ = 2520^\circ \] - **Symmetry:** - **Rotational Symmetry:** 16-fold. - **Reflection Symmetry:** 16 axes. #### **Regular 24-gon (Icositetragon)** **Properties:** - **Interior Angle:** \[ \frac{(24 - 2) \times 180^\circ}{24} = \frac{22 \times 180^\circ}{24} = 165^\circ \] - **Exterior Angle:** \[ \frac{360^\circ}{24} = 15^\circ \] - **Sum of Interior Angles:** \[ (24 - 2) \times 180^\circ = 22 \times 180^\circ = 3960^\circ \] - **Symmetry:** - **Rotational Symmetry:** 24-fold. - **Reflection Symmetry:** 24 axes. --- ### 10) Regular 13-gon (Tridecagon) **Definition:** A tridecagon is a 13-sided polygon. A **regular tridecagon** has all sides of equal length and all interior angles equal. **Properties:** - **Interior Angle:** \[ \frac{(13 - 2) \times 180^\circ}{13} = \frac{11 \times 180^\circ}{13} \approx 152.3077^\circ \] - **Exterior Angle:** \[ \frac{360^\circ}{13} \approx 27.6923^\circ \] - **Sum of Interior Angles:** \[ (13 - 2) \times 180^\circ = 11 \times 180^\circ = 1980^\circ \] - **Symmetry:** - **Rotational Symmetry:** 13-fold. - **Reflection Symmetry:** 13 axes. - **Constructibility:** - A regular polygon is **constructible with compass and straightedge** if the number of sides is a product of distinct Fermat primes and a power of 2. - The number 13 is a Fermat prime (\(2^{2^2} + 1 = 17\)), but since 13 is not a Fermat prime (Fermat primes ≤ 19 are 3, 5, 17, etc.), the regular tridecagon is **not** constructible with compass and straightedge. --- ### Additional Notes on Regular Polygons: 1. **Circumradius (\(R\)) and Inradius (\(r\)):** - **Circumradius** is the radius of the circumscribed circle around the polygon. - **Inradius** is the radius of the inscribed circle within the polygon. - Formulas to relate side length (\(s\)), \(R\), and \(r\): \[ R = \frac{s}{2 \sin\left(\frac{\pi}{n}\right)} \] \[ r = \frac{s}{2 \tan\left(\frac{\pi}{n}\right)} \] 2. **Area Formula:** The area \(A\) of a regular \(n\)-gon with side length \(s\) can be calculated using: \[ A = \frac{n \times s^2}{4 \tan\left(\frac{\pi}{n}\right)} \] 3. **Perimeter (\(P\)):** \[ P = n \times s \] 4. **Symmetry:** Regular polygons exhibit high degrees of symmetry, making them important in various fields such as architecture, art, and molecular chemistry. --- If you have any specific questions or need further details on any of these polygons, feel free to ask!