Calculating the area of a polygon can be simple if it is a figure such as a regular triangle, or very complicated if you are dealing with an irregular shape with eleven sides. If you want to know how to calculate the area of polygons, follow these instructions.
Steps
Part 1 of 3: Finding the Area of a Regular Polygon Using Its Apothem
Step 1. Write the formula to find the area of the regular polygon
It is: area = 1/2 x perimeter x apothem. Here is the meaning of the formula:
- Perimeter: the sum of the lengths of all sides of the polygon.
- Apothem: the segment perpendicular to each side that joins the midpoint with the center of the polygon.
Step 2. Find the apothem of the polygon
If you use the apothem method, its length could be provided in the problem data. Let's say you are calculating the area of a hexagon with an apothem of 10√3.
Step 3. Find the perimeter of the polygon
If this data is provided to you by the problem, then you do not have to do anything else, but it is more likely that you will have to work a little to get it. If you know the apothem and you know that the polygon is regular, there is a way to derive the length of the perimeter. That's how:
- Consider that the apothem is "x√3" of one side of a triangle 30 ° -60 ° -90 °. You can reason in this way because the regular hexagon is made up of six equilateral triangles. The apothem cuts the triangles in half, creating triangles with internal angles of 30 ° -60 ° -90 °.
- You know that the side opposite to the angle of 60 ° is equal to x√3, the side opposite to the angle of 30 ° is equal to x, and that the hypotenuse is equal to 2x. If 10√3 represents "x√3," then x = 10.
- You know that x equals half the length of the base of the triangle. Double it to find the full length. So the base is equal to 20. There are six sides in a regular hexagon, so multiply the length by 20 by 6. The perimeter of the hexagon is 120.
Step 4. Enter the apothem and perimeter values in the formula
The formula you need to use is area = 1/2 x perimeter x apothem, putting 120 in place of the perimeter and 10√3 for the apothem. Here's how it should look:
- area = 1/2 x 120 x 10√3
- area = 60 x 10√3
- area = 600√3
Step 5. Simplify the result
You may be asked to express the result in decimal form instead of the square root. You can use the calculator to find the value of √3 and then multiply it by 600. √3 x 600 = 1, 039.2. This is your final result.
Part 2 of 3: Finding the Area of a Regular Polygon Using Other Formulas
Step 1. Find the area of a regular triangle
To do this you have to follow this formula: area = 1/2 x base x height.
If you have a triangle with a base of 10 and a height of 8, then the area is equal to: 1/2 x 8 x 10 = 40
Step 2. Calculate the area of a square
In this case it is sufficient to raise the length of one side to the second power. It is the same thing as multiplying the base by the height, but since we are in a square where all sides are equal, it means multiplying the side by itself.
If the square has side 6, the area is equal to 6x6 = 36
Step 3. Find the area of a rectangle
In the case of rectangles you have to multiply the base by the height.
If the base is 4 and the height 3, the area will be equal to 4 x 3 = 12
Step 4. Calculate the area of a trapezoid. To find the area of a trapezoid, you must follow the formula: area = [(base 1 + base 2) x height] / 2.
Let's say you have a trapezoid with the bases of 6 and 8 and the height of 10. The area is [(6 + 8) x 10] / 2, simplifying: (14 x 10) / 2 = 70
Part 3 of 3: Finding the Area of an Irregular Polygon
Step 1. Write the coordinates of the vertices of the polygon
The area of an irregular polygon can be obtained by knowing the coordinates of the vertices.
Step 2. Prepare an outline
List the x and y coordinates for each vertex following the counterclockwise order. Repeat the coordinates of the first vertex at the end of the list.
Step 3. Multiply the x coordinate of each vertex by the y coordinate of the next vertex
Add up the results. In this case the sum of the products is 82.
Step 4. Multiply the y coordinate of each vertex by the x coordinate of the next vertex
Once again add up the results. In this case the sum is -38.
Step 5. Subtract the first sum you found from the second
So: 82 - (-38) = 120.
Step 6. Divide the result by 2 and get the area of the polygon
Advice
- If instead of writing the points counterclockwise, you write them clockwise, you will get the value of the area in negative. This can then be a method of identifying the cyclic path or sequence of a given number of points that form a polygon.
- This formula calculates the area with an orientation. If you use it for a figure in which two lines cross as in an eight, you will get the area delimited in an anticlockwise direction minus the area delimited in a clockwise direction.
