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Calculating the area of a polygon is a fundamental concept in geometry. It allows us to determine the amount of space enclosed by a closed figure, whether it’s a simple triangle or a more complex shape with numerous sides and angles. By understanding the techniques and formulas used to calculate the area of different polygons, we can apply this knowledge in real-world scenarios, such as calculating the amount of space within a field or the size of a room. This guide will walk through the step-by-step process of calculating the area of various polygons, providing clear examples and explanations along the way. Whether you’re a student studying geometry or someone seeking to expand your mathematical knowledge, understanding how to calculate the area of a polygon is an essential skill to possess. So let’s delve into this topic, explore the different techniques involved, and discover the beauty behind these geometric calculations.

wikiHow is a “wiki” site, which means that many of the articles here are written by multiple authors. To create this article, 48 people, some of whom are anonymous, have edited and improved the article over time.

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Depending on the shape of the polygon, calculating its area can be simple (as with an equilateral triangle) but can also be very complex (for an irregular 11-sided polygon). This article will help you step by step to know how to calculate the area of different types of polygons.

## Steps

### Calculate the area of a regular polygon using the midpoint

**Formula to calculate the area of a regular polygon.**To calculate the area of a regular polygon, you just need to apply the formula:

*Area = 1/2 x perimeter x median*. The meaning of this formula is as follows:

- Perimeter = sum of the lengths of the sides of the polygon
- Midsegment = line segment perpendicular to the side and passing through the center of the polygon

**Find the midpoint of the polygon.**If you are calculating by the median method, the median length is usually given. For example, consider a regular hexagon with a median length of 10√3.

**Find the perimeter of the polygon.**If the problem is given the perimeter, you just need to assign that value to the expression, but let’s consider the case where we need to find the perimeter. If we already have the length of the median of a regular polygon, we can calculate its perimeter. Here’s how to do it:

- Consider a triangle with three angles of 30-60-90 whose median length is “x√3”. We can do it this way because an equilateral hexagon can be divided into 6 equilateral triangles. The midline of the hexagon will divide each of those equilateral triangles into two triangles with 3 angles of 30-60-90.
- We know that if the side opposite the 30 degree angle has length x, the side opposite the 60 degree angle will have length x√3 and the hypotenuse (the side opposite the 90 degree angle) has a length of 2x. So, in our case, we have 10√3 which is equivalent to “x√3”, so x = 10.
- We already know x = ½ hypotenuse. So the length of the hypotenuse will be 2 times x and equal to 20 units. Since each of the component triangles of the hexagon is an equilateral triangle, the hypotenuse of the triangle 30-60-90 is also the length of the side of the hexagon. A regular hexagon has all 6 sides of equal length, so the perimeter of this hexagon is 20 x 6 = 120.

**Assign the values of the perimeter and the midpoint to the formula.**If you use the above formula:

*Area = 1/2 x perimeter x median*, you just need to substitute 120 for the circumference and 10√3 for the midpoint. We have:

- Area = 1/2 x 120 x 10√3
- Area = 60 x 10√3
- Area = 600√3

**Shorten the results.**You may need to write your answer as a decimal number instead of using a radical sign. In that case, you can simply use a calculator to calculate the value of √3 and multiply it by 600. √3 x 600 = 1039.2. This is the final answer.

### Calculate area of polygon using other formulas

**Calculate the area of an equilateral triangle.**To calculate the area of an equilateral triangle, we use the formula

*Area = 1/2 x base x height.*

- If the triangle has a base of 10 and a height of 8, then the area of the triangle will be 1/2 x 8 x 10, which is 40.

**Calculate the area of the square.**The area of a square is equal to the square of its side lengths. It also means that you multiply the base by the height, and with a square the base and height are the same.

- If the side length of the square is 6 then its area is 6 x 6 = 36.

**Find the area of a rectangle.**The area of a rectangle is equal to the length times the width, or the base times the height.

- If the base of the rectangle is 4 and the height is 3, the area of the rectangle is 4 x 3 = 12.

**Calculate the area of a trapezoid.**The formula for calculating the area of a trapezoid is:

*Area = [(base 1 + base 2) x height]/2*. You can also memorize the following verse: “If you want to calculate the area of a trapezoid – The bottom is big, the bottom is small, we add it – Then multiply it by the height -Divide it by half.”

- Consider a trapezoid with two bases of 6 and 8, respectively, and the height is 10. So the area of this trapezoid is [(6 + 8) x 10]/2, equivalent to [14 x 10]/2, equal to 140/ 2 = 70.

### Calculate the area of an irregular polygon

**Determine the coordinates of the vertices of the irregular polygon.**If we know the coordinates of the vertices of the polygon, we can calculate the area of that polygon.

**Create a table of coordinate values.**List the x, y coordinates of each vertex in counter-clockwise order. Iterate over the first value at the end of the list.

**Multiply the x value of the first vertex in the list by the y value of the second vertex.**Finally add the sum of the products. Considering the example in the figure, we have a sum of 82.

**Multiply the y value of each vertex by the x value of the vertex that follows it.**Adding the products together gives us a value of -38.

**Subtract the second sum from the first sum.**We have 82 – (-38) = 120.

**Divide the obtained difference by 2 to get the area of the polygon.**So we take 120 divided by 2, the result is 60.

## Advice

- If you list the vertices of the polygon in a clockwise order instead of anti-clockwise, the resulting value will be the opposite of the area value to find. Therefore, this way can be used to determine the order of the vertices of a polygon.
- This formula calculates the area of interest in the direction. So, if you apply this formula to a shape that has two intersecting sides like figure 8, you’ll get the counterclockwise area minus the clockwise area.

wikiHow is a “wiki” site, which means that many of the articles here are written by multiple authors. To create this article, 48 people, some of whom are anonymous, have edited and improved the article over time.

This article has been viewed 67,615 times.

Depending on the shape of the polygon, calculating its area can be simple (as with an equilateral triangle) but can also be very complex (for an irregular 11-sided polygon). This article will help you step by step to know how to calculate the area of different types of polygons.

In conclusion, calculating the area of a polygon involves understanding its properties and applying the appropriate formulas. By decomposing the polygon into simpler shapes, such as triangles or rectangles, we can easily find the area of each component and sum them up to obtain the total area of the polygon. Alternatively, the Shoelace Formula provides a more direct method of determining the area by utilizing the coordinates of the polygon’s vertices. It is important to note that the accuracy of the calculations relies on getting accurate measurements and using the correct formulas. Therefore, careful attention should be given to the given dimensions and the mathematical computations involved in calculating the area of a polygon. Overall, knowing how to calculate the area of a polygon is a fundamental skill that is useful in various fields, such as architecture, engineering, and geography.

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