In this section, we delve into the fascinating world of polygons, specifically focusing on regular and irregular polygons including pentagons, hexagons, octagons, and decagons. Polygons form a crucial part of geometry and are defined by their sides, vertices, and interior angles. Understanding their characteristics provides a strong foundation in geometry which is essential for IGCSE students.

Introduction to Polygons

Polygons are flat, two-dimensional shapes consisting of straight lines that are fully closed. The characteristics of polygons vary significantly, depending on the number of sides they have and whether they are considered regular (all angles and sides are equal) or irregular (angles or sides are not equal). This section explores the specific characteristics of pentagons, hexagons, octagons, and decagons.

Regular vs Irregular Polygons

• Regular polygons have all sides of equal length and all interior angles equal.
• Irregular polygons do not have all sides and angles equal.

Image courtesy of Math Monks

Pentagon

A pentagon is a five-sided polygon. It can be regular or irregular.

IImage courtesy of CueMath

Regular Pentagon

• Each internal angle = 108°
• Formula for calculating the area: $A = \frac{1}{4} \sqrt{5(5 + 2\sqrt{5})} s^2$, where $s$ is the side length.

Example

Calculate the area of a regular pentagon with a side length of 6 cm.

Solution:

Hexagon

A hexagon is a six-sided polygon. It is often studied in its regular form.

Regular Hexagon

• Each internal angle = 120°
• Formula for calculating the area: $A = \frac{3\sqrt{3}}{2} s^2$, where $s$ is the side length.

Example

Find the area of a regular hexagon with a side length of 4 cm.

Solution:

Heptagon

A heptagon is a seven-sided polygon that presents a more complex geometric structure compared to polygons with fewer sides.

Regular Heptagon

• Internal angles: Approximately 128.57° each.
• Area formula: $A = \frac{7}{4} s^2 \cot\left(\frac{\pi}{7}\right)$

Example: Area of a Regular Heptagon with Side Length 4 cm

Given: Side length $(s)$ = 4 cm

Solution:

1. Formula: The area formula for a regular heptagon is $A = \frac{7}{4} s^2 \cot\left(\frac{\pi}{7}\right)$.

2. Substitute (s): Plug in $s = 4$ cm into the formula.

3. Calculate: $A = 1.75 \times 4^2 \times \cot\left(\frac{\pi}{7}\right)$

4. Evaluate: $A = 58.14 \, \text{cm}^2$

The area of the regular heptagon, with each side 4 cm long, is approximately $58.14 \, \text{cm}^2$, after evaluating the cotangent function and completing the multiplication.

Octagon

An octagon is an eight-sided polygon. The regular octagon is most commonly referenced.

Regular Octagon

• Each internal angle = 135°
• Formula for calculating the area: $A = 2(1+\sqrt{2})s^2$, where $s$ is the side length.

Example

Calculate the area of a regular octagon with a side length of 5 cm.

Solution:

Nonagon

A nonagon, also known as an enneagon, is a nine-sided polygon that offers a rich field for geometric exploration.

Regular Nonagon

• Internal angles: Approximately 140° each.
• Area formula: $A = \frac{9}{4} s^2 \cot\left(\frac{\pi}{9}\right)$

Example: Area of a Regular Nonagon with Side Length 3 cm

Given: Side length $(s)$ = 3 cm

Solution:

1. Formula: The area formula for a regular nonagon is $A = \frac{9}{4} s^2 \cot\left(\frac{\pi}{9}\right)$.

2. Substitute (s): Plug in $s = 3$ cm into the formula.

3. Calculate: $A = 2.25 \times 3^2 \times \cot\left(\frac{\pi}{9}\right)$

4. Evaluate: $A \approx 55.64 \, \text{cm}^2$

The area of the regular nonagon, with each side 3 cm long, is approximately $55.64 \, \text{cm}^2$, showcasing the application of trigonometry and geometric formulas to determine the area of polygons with a higher number of sides.

Decagon

A decagon is a ten-sided polygon, with the regular form being the primary focus of study.

Regular Decagon

• Each internal angle = 144°
• Formula for calculating the area: $A = \frac{5}{2} s^2 \sqrt{5 + 2\sqrt{5}}$, where $s$ is the side length.

Example

What is the area of a regular decagon with a side length of 3 cm?

Solution:

$A = \frac{5}{2} \times 3^2 \sqrt{5 + 2\sqrt{5}} \approx 84.30 \, \text{cm}^2$

Practice Question

Given a regular hexagon and a regular decagon that have the same perimeter, find the ratio of their areas.

Solution:

Let the side length of the hexagon be $h$ and the decagon be $d$. Since they have the same perimeter:

$6h = 10d \Rightarrow h = \frac{5}{3}d$

$\text{ Area of the hexagon } = \frac{3\sqrt{3}}{2} h^2$

$\text{ Area of the decagon } = \frac{5}{2} d^2 \sqrt{5 + 2\sqrt{5}}$

Substitute $h = \frac{5}{3}d$ into the hexagon's area formula and compare.