Polygons are a fundamental feature of geometry. From the ancient pyramids to modern architecture, polygons have been pivotal in shaping our environment. Put simply, polygons are flat shapes of straight lines that are joined to form a closed circuit. An example is a triangle - the polygon with the fewest sides - which can be extended to a shape with limitless sides. The beauty of polygons lies in their variety and properties, which are foundational to understanding more complex geometric and mathematical concepts.
This article will guide you through the basics of polygons, distinguishing between regular and irregular types and introducing the concepts of convex, concave and complex polygons. Examples and quizzes are included to help you understand these versatile shapes that are a topic in the GCSE Maths curriculum.
Types of polygons
Polygons aren't just confined to the basic shapes we encounter in everyday life; they span a vast spectrum of forms, each with unique properties and classifications. This section will discuss the various types of regular polygons, from the most simplistic to the most complex.
Regular vs. Irregular Polygons
As part of the GCSE Maths curriculum, you'll need to differentiate between a regular or irregular polygon. Both polygon types have been summarised below.
A regular polygon is a figure with all sides of equal length and all interior angles of equal measure. The symmetry and uniformity of regular polygons make them aesthetically pleasing and mathematically significant. For example, a triangle or square polygon is regular.
Irregular polygons lack uniform side lengths and angle measures. Shapes like rectangles (which have equal opposite sides but differ in length from adjacent sides) and trapezoids (with only one pair of parallel sides) fall into this category. Despite their lack of uniformity, irregular polygons are important, offering a broader understanding of geometric principles and real-world applications.
Convex and Concave Polygons
A convex polygon has all interior angles less than 180°. This means no interior angle pushes "inwards" towards the interior of the polygon. Convex polygons are the "outward" bulging figures, simple yet powerful in their properties.
Concave polygons have at least one interior angle greater than 180°, giving the appearance of being "caved in" or having a dent. This classification includes shapes like star polygons and arrows, where parts of the boundary push into the figure.
Understanding the difference between convex and concave polygons is needed for mathematical and practical tasks, from computing areas to solving complex geometric problems.
Simple vs. Complex Polygons
A simple polygon has sides that only meet at their endpoints, never crossing over each other. You will mainly learn about simple polygons in Maths GCSE, such as pentagons and hexagons, which are straightforward in their construction.
Complex polygons have sides that intersect, creating a shape that cannot be considered a single, continuous boundary. The most common examples are star shapes, like the pentagram, which can be thought of as a pentagon with additional lines drawn between non-adjacent vertices to form a star.
1
Which of the following is a regular polygon?
Angles in a Regular Polygon
The concept of angles plays a role in developing our understanding of polygons. Every polygon is defined not just by its sides but also by its angles. In this section, we'll learn about interior and exterior angles, discussing how they define polygons and their properties.
Interior Angles formula
The interior angles of a polygon are the angles found at each vertex on the inside of the polygon. One of the most interesting aspects of interior angles is their relationship with the number of sides in a polygon.
The sum of the interior angles of a polygon can understood by this formula:
Sum of interior angles = (n-2)×180o
Here, n represents the number of sides. This formula reveals that as the number of sides increases, so does the sum of the interior angles. For regular polygons, where all sides and angles are equal, finding the measure of each interior angle becomes straightforward once the sum of all interior angles is known.
Exterior Angles
Exterior angles exist at each vertex outside the polygon, formed by extending one side of the polygon. Curiously, the sum of the exterior angles of a polygon is always 360°. This constant provides a solid foundation for solving many geometric problems.
For a regular polygon, calculating the size of an exterior angle is simple with the formula:
Size of an exterior angle = n360°
Here, n represents the number of sides. This equation shows the relationship between the number of sides and the size of each exterior angle.
2
How much is each interior angle of a regular octagon?
Perimeters and areas of regular Polygons
Unraveling the boundaries and spaces within
Now we have covered the angles that shape polygons, we can look at their physical dimensions: perimeters and areas. These measures define the extent of polygons and unlock practical applications ranging from architecture to land surveying. This section will enable you to calculate the perimeter and area of various polygons.
Perimeters of Polygons
The perimeter of a polygon is the total length around its boundary. Calculating the perimeter is straightforward for regular polygons with this formula since all sides are equal:
Perimeter = number of sides × length of one side
For irregular polygons, the perimeter is the sum of the lengths of all sides. Understanding how to calculate perimeters is crucial, as it forms the basis for more complex geometric and practical calculations.
Areas of Polygons
The area measures the space inside a polygon. The formulas for calculating the area vary depending on the type of polygon, reflecting their unique properties:
- For regular polygons (like squares and equilateral triangles), specific formulas allow for direct calculation.
- For rectangles and parallelograms, the area is calculated as the product of base and height.
- Triangles have a unique formula: Area = 12 × base × height.
- More complex polygons may require you to divide the shape into simpler components or use trigonometric methods.
This variability shows how principles in maths can be adapted to understand and work with different geometric shapes.
3
How do you calculate the area of an irregular pentagon?
Polygons examples in the Real World
The geometry of polygons extends into the world around us. This section will explore how polygons serve as the building blocks in various fields, from architecture and art to technology and nature, demonstrating their relevance beyond the curriculum.
Architecture and design
Polygons are the cornerstone of architectural design and construction. Regular polygons often form the basis of aesthetically pleasing structures. For instance, hexagonal patterns are common in tiling designs due to their efficiency in covering a surface without gaps. The geodesic domes, comprising triangles, hexagons and pentagons, highlight the strength and stability polygons can offer to large-scale structures.
Art and decoration
The use of polygons in art dates back to ancient times, where they appear in artistic motifs and designs. In modern times, polygons inspire both digital and physical art forms. Tessellations, which are patterns made of polygons that fit together without gaps or overlaps, are a common theme in decorative arts. The work of M.C. Escher exemplifies the artistic exploration of polygons and their transformation into complex, mesmerising patterns.
Technology and computer graphics
Polygons are fundamental in computer graphics because they render three-dimensional objects on a two-dimensional screen. The complexity of an object’s surface can be represented by a mesh of polygons, allowing for detailed and realistic visuals in video games and simulations. The study of polygons is also important in developing algorithms to optimise rendering efficiency and visual fidelity.
For further information, Khan Academy offers an engaging course on computer science and programming, featuring modules on graphics and animation.
Nature and environmental planning
Polygons are seen naturally in the environment. The hexagonal columns of the Giant’s Causeway or the tessellated pavement are geological formations that showcase polygons in nature. Understanding the geometry of these shapes helps in environmental planning and conservation efforts, allowing for the mapping of natural terrains and the efficient allocation of land resources.
4
Why are hexagons frequently used in architectural tiling?
Final thoughts
Polygons are commonly used in architecture, art, technology and nature. They are applied both for functional and aesthetic purposes. This usage highlights the significance of understanding polygons for passing exams and navigating the world more creatively. With their simple beginnings and complex implications, polygons symbolise the beauty and utility of mathematics.
As you continue to learn about geometry, consider using tools like GeoGebra This is interactive mathematics software that allows dynamic exploration of geometric figures, including polygons. TeachTutti provides a list of Maths GCSE tutors if you need support revising this topic. These qualified tutors can provide personalised support, helping students to achieve their academic goals.
Glossary
- Polygon - A flat, two-dimensional shape with straight sides that are fully closed. The sides do not cross each other.
- Regular Polygon - A polygon where all sides and all angles are equal in length and measure, respectively.
- Irregular Polygon - A polygon that does not have all sides of equal length and all angles of equal measure.
- Convex Polygon - A polygon where no line segment between any two points in the shape passes outside of the polygon. All interior angles are less than 180 degrees.
- Concave Polygon - A polygon that has at least one interior angle greater than 180 degrees, causing the polygon to appear as if it is "caved in" or has a dent.
- Simple Polygon - A polygon whose sides intersect only at their endpoints, with no sides crossing over each other.
- Complex Polygon - A polygon that has sides that intersect or cross over each other at points other than the endpoints.
- Interior Angle - An angle inside a polygon formed by two adjacent sides.
- Exterior Angle - An angle formed by one side of a polygon and the extension of an adjacent side. The sum of the exterior angles of any polygon is 360 degrees.
- Perimeter - The total length of all the sides of a polygon.
- Area - The amount of space inside the boundary of a polygon, measured in square units.
- Tessellation - A pattern made of polygons that fit together perfectly without gaps or overlaps, covering a surface.
- Mesh - A collection of polygons that model the surface of a three-dimensional object in computer graphics. They are often triangles.
- Geodesic Dome - A spherical structure made from a network of polygons that distribute stress evenly across the structure, making it extremely strong and lightweight. They are often triangles.
