Polygons

---
config:
  look: handDrawn
  theme: dark
---

graph TD;
  Rc@{ shape: rect, label: "Rectangle" }
  style Rc fill:#555, stroke:#AAA, stroke-width:2px
  Tr@{ shape: tri, label: "Triangle" }
  style Tr fill:#559, stroke:#AAF, stroke-width:2px
  Trap@{ shape: trap-t, label: "Trapezoid" }
  style Trap fill:#595, stroke:#AFA, stroke-width:2px
  hex@{ shape: hex, label: "Hexagon" }
  style hex fill:#599, stroke:#AFF, stroke-width:2px
  circle@{ shape: circle, label: "Circle" }
  style circle fill:#955, stroke:#FAA, stroke-width:2px
  diamond@{ shape: diamond, label: "Square" }
  style diamond fill:#959, stroke:#FAF, stroke-width:2px

  Rc ~~~ circle
  Tr ~~~ hex
  Trap ~~~ diamond

A polygon is a closed, two-dimensional shape formed by straight line segments. Each polygon has a specific number of sides, vertices, and angles. The simplest polygons include triangles, quadrilaterals, pentagons, and so on.

Polygons can be classified based on their number of sides:

• Triangle (3 sides)
• Quadrilateral (4 sides)
• Pentagon (5 sides)
• Hexagon (6 sides)
• Heptagon (7 sides)
• Octagon (8 sides)
• ... and so on

Polygons can also be regular (all sides and angles are equal) or irregular (sides and angles vary in length and measure).

Paralelogram

A parallelogram is a specific type of quadrilateral with two pairs of parallel sides. It has the following properties:

• Opposite sides are equal in length.
• Opposite angles are equal.
• The diagonals bisect each other at their midpoint.

📘 Perimeters of Polygons

The perimeter of a polygon is the total length of its sides. It is calculated as:



\color{#995}
P = ∑ \text{side lengths}

For specific polygons:

• Triangle: $P = a + b + c$

• Square: $P = 4s$ (where $s$ is the side length)

• Rectangle: $P = 2(l+w)$ (where $l$ is the length and $w$ is the width)

• Regular Polygon (with $n$ sides of length $s$ ): $P = n⨯s$

Understanding perimeters is useful in real-world applications such as fencing, tiling, and geometry-based engineering projects.

Area

The area of a polygon represents the amount of surface it covers. It is measured in square units, such as square meters ($m^2$) or square centimeters ($cm^2$). It reflects the size of a surface.

For specific polygons:

• Rectangle: $$A = l \times w$$
• Square: $$A = s^2$$
• Triangle: $$A = \frac{1}{2} b h$$
• Parallelogram: $$A = b h$$
• Regular Polygon (with apothem $$a$$ and perimeter $$P$$): $$A = \frac{1}{2} a P$$

Understanding area calculations is essential for construction, land measurement, and various engineering applications.

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Polygons are fundamental in mathematics and practical applications. From architectural designs to scientific modeling, their properties provide essential tools for measurement and problem-solving. By exploring different polygons, their perimeters, and their special cases like parallelograms, we unlock deeper insights into the geometry of the world around us.