Area of Polygons – Formula, Definition With Examples

Welcome to the fantastic world of shapes, lines, and angles, where geometry rules the roost! We, at Brighterly, believe that the journey of learning should be exciting, engaging, and illuminating. Today, we dive into the heart of geometry – polygons, and specifically, the area of polygons. Our journey will take us through different types of polygons – regular and irregular, we will decipher their unique properties, and most importantly, learn how to calculate their areas. As with all our lessons, this one too is packed with illustrative examples, hands-on practice problems, and detailed explanations that will help our young learners strengthen their foundational understanding of geometry. So buckle up, bring along your protractor, and let’s set sail into the world of polygons!

What Are Polygons? – Definition With Examples

Polygons are 2-dimensional figures formed by a sequence of straight lines, also known as edges or sides, that close in a loop to form a closed chain or circuit. They are among the earliest and most common shapes studied in geometry, dating back to the times of ancient Greece. Simple examples include shapes like a triangle, quadrilateral (square, rectangle), pentagon, hexagon, and so on. The vertices or corners are the points where the edges of a polygon meet. To get a more visual grasp of polygons, you can refer to the polygon page on our Brighterly website.

Definition of Regular Polygons

In the world of polygons, regular polygons hold a special place. A regular polygon is a polygon that is both ‘equiangular’ (all angles are equal in measure) and ‘equilateral’ (all sides have the same length). Think of a perfect square or an equilateral triangle. These polygons are regular because each of their angles and sides are equal.

Definition of Irregular Polygons

Contrary to regular polygons, irregular polygons don’t have sides and angles of equal measure. They can come in a variety of unique and interesting shapes and sizes, like a scalene triangle or a trapezium. Irregular polygons are often found in real-life objects, giving them a touch of realism. They can be more complex to analyze compared to regular polygons due to their lack of symmetry.

Understanding the Area of Polygons

Area, in the simplest terms, refers to the amount of space a shape occupies. In the context of polygons, the area of a polygon is calculated based on the number of square units it covers. This concept is crucial in understanding and comparing different polygons, and it has wide applications in fields like architecture, design, cartography, and more. Understanding area is not just about solving mathematical problems, it’s about making sense of the space around us.

Area Formulas for Different Types of Polygons

There are specific formulas to calculate the area of different types of polygons. For example, the area of a rectangle is length times width, while for a triangle it’s one half of the base times the height. These formulas provide a systematic approach to determining the space occupied by these shapes.

Area Formula for Regular Polygons

In case of regular polygons, the formula for area varies depending on the number of sides. For example, the area of a regular triangle (equilateral) is √3/4 * (side length)^2, while for a square it is (side length)^2. We have an in-depth exploration of regular polygon area formulas available on Brighterly.

Area Formula for Irregular Polygons

Calculating the area of irregular polygons can be more complex. Often, they are divided into simpler shapes (like triangles and rectangles), and the areas of these shapes are calculated individually and summed. This process is also known as tessellation. To find out more about calculating areas of irregular polygons, visit our specific page dedicated to it.

Properties of Area in Polygons

Understanding the properties of area in polygons allows us to draw meaningful conclusions about these shapes. For instance, if all sides of a polygon increase by a certain factor, its area will increase by the square of that factor. Or, among polygons of the same number of sides, the regular polygon has the maximum area.

Properties of Area in Regular Polygons

Just as with polygons in general, regular polygons have their own unique set of properties when it comes to their areas. One important property to note is that for a given perimeter, a regular polygon will always have a larger area than any irregular polygon. For more details, refer to our page on the properties of area in regular polygons.

Properties of Area in Irregular Polygons

The properties of area in irregular polygons can be quite varied due to their inherent lack of symmetry. However, one common feature they share is that they always have smaller areas than regular polygons of the same perimeter.

Difference Between Regular and Irregular Polygons

While regular and irregular polygons can both provide interesting challenges in understanding their properties, it’s important to remember the key differences. Regular polygons are symmetrical, with all sides and angles equal, whereas irregular polygons lack this symmetry. This difference greatly affects the calculation of their areas. For more information, see our comparison of regular and irregular polygons.

Calculating Area of Polygons

Learning how to calculate the area of polygons is a crucial part of geometry education. Equipped with the right formula and approach, you can calculate the area of any polygon, whether it’s regular or irregular.

In geometry, each type of polygon has its own specific formula to find the area. For instance, the area of a rectangle is determined by multiplying its length and width, whereas, for a triangle, you take half of the base length and multiply it by the height.

Moreover, the approach for calculating the area can vary based on the type of polygon – regular or irregular. Regular polygons have equal sides and angles, allowing for uniformity in the calculation. Irregular polygons, on the other hand, can have varying side lengths and angles, making the process of calculation slightly more complex.

Steps to Calculate Area of Regular Polygons

Calculating the area of regular polygons involves specific steps that can be easily followed once understood. Here’s a general guide:

  1. Identify the type of regular polygon: Knowing whether you’re dealing with a square, an equilateral triangle, or a regular pentagon, for instance, is essential because the formula to calculate the area depends on the type of polygon.

  2. Apply the appropriate area formula: Once you’ve identified the type of regular polygon, it’s time to apply the corresponding formula.

  • Square: side length^2
  • Equilateral Triangle: (sqrt(3) / 4) * (side length)^2
  • Regular Pentagon: (1 / 4) * sqrt(5 * (5 + 2 * sqrt(5))) * (side length)^2
  1. Perform the calculation: Once the appropriate formula is applied, perform the calculation to find the area. Don’t forget to double-check your calculations for accuracy.

Steps to Calculate Area of Irregular Polygons

The process to calculate the area of irregular polygons can be more complex, but it’s achievable. Typically, the approach involves dividing the irregular polygon into simpler, regular polygons like rectangles and triangles, for which we know how to calculate the area. Here are the steps:

  1. Divide the irregular polygon into regular polygons: This can often be achieved by drawing lines from the vertices to create triangles or rectangles within the polygon.

  2. Calculate the area of each regular polygon: Using the formulas mentioned above, calculate the area of each smaller regular polygon.

  3. Sum up the areas: Add up all the individual areas to get the total area of the irregular polygon.

Practice Problems on Area of Polygons

Practicing calculations can help cement your understanding. Here are some practice problems on the area of polygons:

  1. Rectangle: A rectangle has a length of 6 cm and a width of 4 cm. What’s its area? Solution: Area = length * width = 6 cm * 4 cm = 24 square cm.

  2. Equilateral Triangle: An equilateral triangle has a side length of 5 cm. What’s its area? Solution: Area = (sqrt(3) / 4) * (side length)^2 = (sqrt(3) / 4) * (5 cm)^2 = 10.83 square cm.

  3. Irregular Polygon: An irregular quadrilateral can be divided into a rectangle of 6 cm by 4 cm and a right triangle with a base of 4 cm and height of 3 cm. What’s its area? Solution: Area = area of rectangle + area of triangle = (6 cm * 4 cm) + (1/2 * 4 cm * 3 cm) = 24 cm^2 + 6 cm^2 = 30 square cm.

Information Sources
  1. Wolfram MathWorld: Polygons
  2. US Department of Education: Mathematics
  3. Wikipedia: Polygon

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