Calculating the area of a polygon can be as simple as finding the area of a regular triangle or as complex as finding the area of eight irregular areas. If you want to know how to find the area of a polygon, follow these steps:
Step
Method 1 of 3: Finding the Area of a Polygon Using the Apothem
Step 1. Write down the formula to find the area of the polygon
To find the area of a regular polygon, all you need to do is follow this simple formula: Area = 1/2 x side length x apothem. Here is what it means:
- Side length = sum of the lengths of all sides
- Apothem = perpendicular line connecting the center of the polygon to the midpoint of any side.
Step 2. Find the apothem of the polygon
If you use the apothem method, then the apothem must be available to you. Let's say you are looking for the area of a hexagonal plane that has an apothem length of 10√3.
Step 3. Find the length of the side of the polygon
If you've found the side lengths, then you're almost done, but there's probably still something you need to do. If the apothem value is available for a regular polygon then you can use it to find the side lengths. Here's how:
- Think of the value of the apothem as the "x√3" value of a 30-60-90 degree triangle. You can estimate this value because the hexagon is made up of six equal triangles. The apothem will divide the plane into two equal planes, thus creating a triangle with an angle measuring 30-60-90 degrees.
- You know that the side opposite the 60 degree angle has length = x√3, so the side opposite the 30 degree angle will have length = x, and the side opposite the 90 degree angle will have length = 2x. If 10√3 represents "x√3," then the value of x = 10.
- You know that x = half the length of the bottom side of the triangle. Double the value to get the full length. So the length of the whole triangle is 20. There are six of these sides in a hexagon, so multiply by 20 x 6 to get the side length of the hexagonal 120.
Step 4. Plug the apothem value into the formula
If you use the formula Area = 1/2 x side length x apothem, then you can enter 120 as the side length and 10√3 as the apothem value. Then the formula will look like this:
- Area = 1/2 x 120 x 10√3
- Area = 60 x 10√3
- Area = 600√3
Step 5. Simplify your answer
You may need to express yours in decimal numbers and not in square root values. Use your calculator to find the closest value to 3 and multiply by 600. 3 x 600 = 1.039, 2. This is your final answer.
Method 2 of 3: Finding the Area of a Polygon Using Other Formulas
Step 1. Find the area of a regular triangle
If you want to find the area of a regular triangle, all you have to do is 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 Area = 1/2 x 8 x 10, or 40
Step 2. Find the area of the square
To find the area of a square, multiply both sides. This is the same as multiplying the base by the height of a square, because the base and height are the same.
If the square has 6 sides, then its area is 6 x 6, or 36
Step 3. Find the area of the rectangle
To find the area of a rectangle, multiply the length by the width.
If the length of the rectangle is 4 and the width is 3, then the area of the rectangle is 4 x 3, or 12
Step 4. Find the area of the trapezoid
To find the area of a trapezoid, you need to follow the following formula: Area = [(base 1 + base 2) x height]/2.
Let's say you have a trapezoid with bases 6 and 8 and a height of 10. Then the area is [(6 + 8) x 10]/2, which can be simplified to (14 x 10)/2, or 140/2, so the area is 70
Method 3 of 3: Finding the Area of an Irregular Polygon
Step 1. Write down the coordinates of the irregular polygon
It is possible to determine the area of an irregular polygon if you know the coordinates of each corner.
Step 2. Create a collation list
Write down the x and y coordinates of each corner of the polygon in a counterclockwise direction. Repeat the coordinates of the first point at the bottom of your list.
Step 3. Multiply the x-coordinate value of each point by the y-value of the next point
Add up the results, which is 82.
Step 4. Multiply the y-value of each point's coordinates by the x-value of the next point
Similarly, add up the results. The total value in this example is -38.
Step 5. Subtract the second value from the first value
Subtract -38 from 82 so that 82 - (-38) = 120.
Step 6. Divide these two increment values to get the area of the polygon
Divide 120 by 2 to get 60 and you're done.
Tips
- If you write the dot list clockwise then you will get a negative area value. Thus, this method can be used to check the order of the list of points that make up the polygon.
- This formula can calculate the area with a certain direction. If you use it on a plane where the two lines intersect like a figure eight, you'll get the area around it minus the area clockwise.
