Area of a Polygon Definitions and Examples
Introduction
In geometry, an area is the 2-dimensional space or region occupied by a closed figure, while a polygon is a plane figure that is “bounded by straight line segments”. Both of these concepts are important when trying to calculate the size or dimensions of an object. In this blog post, we will explore the concept of area in more depth and provide some examples of how to calculate the area of a polygon. We will also discuss some of the applications of this concept in real-world scenarios. So if you’re ready to learn more about area, then read on!
Area of Polygons
The area of a polygon is the two-dimensional space enclosed by the sides of the polygon. The surface area of a three-dimensional object is the sum of the areas of its faces.
In mathematical terms, the area of a closed figure is the number of unit squares that cover the surface of the figure. The most common unit for measuring area is the square inch, but other units such as square feet or square yards can also be used.
To find the area of a regular polygon, one can use the formula:
Area = 1/2 * perimeter * apothem
What is the Area of a Polygon?
A polygon is a closed figure made up of line segments. The area of a polygon is the number of square units inside the polygon. The most basic polygons are triangles and quadrilaterals.
To find the area of a polygon, we can use the formula:
Area = 1/2 * base * height
This formula only works for regular polygons, which are polygons with all sides and angles equal. To find the area of an irregular polygon, we can divide it into regular shapes and then add up the areas of those shapes.
Difference Between Perimeter and Area of Polygons
When it comes to polygons, perimeter is the distance around the outside of the shape while area is the measure of the total surface inside the shape. In other words, perimeter is a one-dimensional measurement (line) and area is a two-dimensional measurement (surface).
To better understand the difference between perimeter and area, let’s look at a few examples:
A square has a perimeter of 4 sides x side length. So, if each side is 3 feet long, then the square’s perimeter would be 12 feet. The formula for finding the area of a square is side length x side length. So using our 3 foot sides, we would calculate 3 feet x 3 feet = 9 square feet.
A rectangle has a perimeter of 2 lengths x width + 2 widths x length. So if our rectangle was 3 feet long and 5 feet wide, then its perimeter would be 2(3) + 2(5) = 16 feet. The formula for finding the area of a rectangle is length x width, so our calculation would be 3 feet x 5 feet = 15 square feet.
A triangle has a perimeter of base + 2 sides. So if our triangle had a base that was 6 inches long and sides that were 4 inches long, then its perimeter would be 6 + (2)(4) = 14 inches. The formula for finding the area of a triangle is 1/2 base x height. So our triangle with a 6 inch base and
Area of Polygon Formulas
There are a few different formulas that can be used to calculate the area of a polygon, depending on the shape of the polygon and the information that is known.
If the polygon is a regular polygon, then the area can be calculated using the following formula: A = (1/2)*n*s^2, where n is the number of sides and s is the length of each side.
If the polygon is an irregular polygon, then the area can be calculated by dividing it into smaller triangles and adding up the areas of those triangles. The formula for calculating the area of a triangle is: A = (1/2)*b*h, where b is the base and h is the height.
Another way to calculate the area of an irregular polygon is to use its centroid. The centroid is simply the average of all of the vertices (x coordinates) divided by 2, and the average of all of y coordinates divided by 2. Once you have found these averages, plug them into this formula: A = 1/2 *abs(det(A)), where A=(x_1,…x_n|y_1,…y_n). This formula might look daunting, but it’s not too bad once you break it down!
Area of Regular Polygons
A regular polygon is a polygon that is both equiangular and equilateral. Equiangular means that all of the polygon’s angles are equal. Equilateral means that all of the sides of the polygon are the same length. Because all sides and angles of a regular polygon are equal, each regular polygon is a simple figure.
There are two ways to find the area of a regular polygon. The first way is to use the formula:
A = (1/2) * n * s^2
where A is the area, n is the number of sides, and s is the length of each side. The second way to find the area of a regular polygon is to use the formula:
A = apothem * perimeter
where A is the area, apothem is the distance from the center to one of the vertices, and perimeter is the distance around the entire figure.
Area of Irregular Polygons
The area of an irregular polygon can be defined as the measure of the region enclosed by the polygon. It is often calculated by dividing the polygon into smaller, more manageable pieces and then adding up the areas of those individual pieces.
There are a few different methods that can be used to calculate the area of an irregular polygon. The most common method is to divide the polygon into triangles, then use the formula for the area of a triangle to calculate the area of each individual triangle. Once you have the areas of all the triangles, you can add them up to get the total area of the polygon.
Another method that can be used is to divide the polygon into quadrilaterals, then use the formula for the area of a quadrilateral to calculate the area of each individual quadrilateral.
Area of Polygons with Coordinates
We can calculate the area of a polygon by adding together the areas of the triangles that make it up. If we know the coordinates of the vertices of the polygon, we can use a formula to calculate the area.
The area of a triangle with vertices at (x1,y1), (x2,y2), and (x3,y3) is
| x1 y2 x2 y3 x3 y1 |
A= ——————
2
So, if we have a polygon with n vertices at (xi,yi) for i = 1,…,n then the area A is given by
A = 1/2 * sum_{i=1}^{n-1} (x_i y_{i+1} – x_{i+1} y_i)
Area of Polygons Examples
There are many shapes that fall under the category of polygons, and each one has a different formula for calculating its area. To find the area of a polygon, you need to know the length of each side and the measurement of the angles between each side. Once you have this information, you can plug it into one of the formulas below to find the area.
– Rectangles: A = lw (area = length x width)
– Squares: A = s^2 (area = side squared)
– Triangles: A = 1/2bh (area = 1/2 base x height)
– Regular polygons: A = ap/2 (area = apothem x perimeter/2)
– Irregular polygons: A = ab/sin(C) (area = base x height/sin(central angle))
Practice Questions on Area of Polygons
When finding the area of a polygon, you will need to use a different formula depending on whether the polygon is regular or irregular. A regular polygon is a polygon where all sides and angles are equal, while an irregular polygon is a polygon that has different side lengths and/or angles.
To find the area of a regular polygon, you will use the formula:
Area = (n * s^2) / (4 * tan(180/n))
where n is the number of sides and s is the length of each side.
To find the area of an irregular polygon, you will first need to divide the polygon into smaller, regular polygons. You can then find the area of each small polygon and add them together to get the total area of the irregular polygon.
FAQs on Area of Polygons
- What is the area of a polygon?The area of a polygon is the amount of two-dimensional space that the figure occupies. The simplest way to calculate the area of a polygon is to use the formula: A = 1/2 * b * h, where b is the base and h is the height. However, this formula only works for certain polygons, such as triangles and rectangles. More complicated polygons will require different formulas.2. How do you find the area of a irregular polygon?
There are a few different ways to find the area of an irregular polygon. One method is to divide the polygon into smaller, more manageable shapes, such as triangles and rectangles, and then use the appropriate formula for each shape. Another method is to use integration, which can be used to find the area under a curve that represents the outline of the polygon.
3. What are some examples of polygons?
Some common examples of polygons include triangles, squares, rectangles, pentagons, and hexagons. These figures all have straight sides and can be classified based on the number of sides they have. Polygons can also have more than six sides, but they are generally less common and more difficult to work with mathematically.
Conclusion
We hope this article has helped you understand the concept of area for a polygon and given you some examples to work with. Remember, the formula for finding the area of a regular polygon is: A = 1/2 * ap, where a is the length of one side and p is the perimeter. If you need help figuring out the lengths or perimeters of polygons, there are many online calculators that can assist you. Have fun exploring the world of geometry!
