Calculating the area of a polygon can be as simple as figuring out the area of a triangle or as complicated as figuring out the area of an eleven-sided irregular figure. To learn how to calculate the area of a variety of polygons, check out the following article.
Method 1 of 3: Regular Polygons
Step 1. Use the default formula for all regular polygons
The simple formula to find the area of a regular polygon (with all sides and all angles equal) is: area = 1/2 x perimeter x apothema. In other words, this formula means that:
- Perimeter = the sum of the length of all sides.
- Apothema = a part that joins the center of the polygon to the middle of whatever side is perpendicular.
Step 2. Discover the apothema of the polygon
If you are using the apothema method, the value will be given to you. For example, let's work with a hexagon that is 10√3 long.
Step 3. Find the perimeter of the polygon
If the perimeter value is given to you, then the job is almost done. If the value of the apothema is also known and you are working with a regular polygon, use the apothema to calculate the perimeter. Here's the step by step:
- Think of the apothema as the "x√3" side of a triangle with 30-60-90 degrees. You can visualize it this way because the hexagon is made up of six equilateral triangles. The apothema cuts them in half, forming a triangle with angles of 30-60-90 degrees.
- You know that the side opposite the 60-degree angle is = x√3, the side opposite the 30-degree angle is = x, and the side opposite the 90-degree angle is = 2x. If 10√3 represents "x√3", then it can be concluded that x = 10.
- You know that x = half the length of the underside of the triangle. Double its value to get the full length. The underside of the triangle is 20 units long. There are six of these sides to the hexagon. Then multiply 20 x 6 to get 120, the perimeter of the hexagon.
Step 4. Insert the value of the apothema and the perimeter into the formula
If you are using the formula area = 1/2 x perimeter x apothema, " then you can fit 120 for the perimeter and 10√3 for the apothema. Here's the example:
- area = 1/2 x 120 x 10√3.
- area = 60 x 10√3.
- area = 600√3.
Step 5. Simplify your answer
It may be necessary to give the result in decimals instead of leaving it as a square root. Use the calculator to get the closest match for √3 and then multiply the result by 600. √3 x 600 = 1,039, 2. This is the final result.
Method 2 of 3: Calculating the Area of Regular Polygons Using Other Formulas
Step 1. Calculate the area of a regular triangle
Just use the following formula: area = 1/2 x base x height.
For example, if your triangle is 10 base and 8 high, then the area is equal to = 1/2 x 8 x 10, that is, 40
Step 2. Calculate the area of a square
Just square either side. It would be the same as multiplying the base by the height, since they are equal in square.
For example, if the square is 6 on its side, then the area is equal to 6 x 6, that is, 36
Step 3. Calculate the area of a rectangle
Just multiply the base by the height.
For example, if the base of the rectangle is 4 and the height is 3, then the area is equal to 4 x 3, ie 12
Step 4. Calculate the area of a trapeze
Just follow this formula: area = [(base 1 + base 2) x height]/2.
For example, imagine a trapeze with bases equal to 6 and 8 and a height of 10. Applying the formula, we have [(6 + 8) x 10]/2, which can be simplified to (14 x 10)/2, or 140/2, which results in an area equal to 70
Method 3 of 3: Calculating the Area of Irregular Polygons
Step 1. Note the coordinates at the vertices of the irregular polygon
To determine the area of an irregular polygon, it is very useful to know the coordinates of the vertices.
Step 2. Make a vector
List the x and y coordinates of each polygon vertex counterclockwise. Repeat the coordinates of the first point at the end of the list.
Step 3. Multiply the x coordinate of each vertex by the y coordinate of each vertex
Add up the results. The total of products is 82.
Step 4. Multiply the y coordinate of each vertex by the x coordinate of the next vertex
Add up the results. The sum total of these results is -38.
Step 5. Subtract the sum of the first products from the sum of the second products
Subtract -38 from 82 to get 82 - (-38) = 120.
Step 6. Divide the difference by 2 to get the polygon area
Just divide 120 by 2 to get 60. Mission accomplished!
- If you list the points clockwise instead of counterclockwise, you will have the area in a negative number. So, this can be used as a tool to identify a cyclical or sequential path of a given set of points forming a polygon.
- This formula computes area with orientation. If you use it in a format where two lines intersect like an 8, you will have the counterclockwise enclosed area minus the clockwise enclosed area.