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Calculating the area of a polygon can be as simple as finding the area of a regular triangle or as complicated as finding the area of an irregular elevensided shape. If you want to know how to find the area of a variety of polygons, just follow these steps.
Steps
Area Help
Part 1
Part 1 of 3:
Find the Area of Regular Polygons Using Their Apothems

1Write down the formula for finding the area of a regular polygon. To find the area of a regular polygon, all you have to do is follow this simple formula: area = 1/2 x perimeter x apothem.^{[1] X Research source } Here is what it means:
 Perimeter = the sum of the lengths of all the sides
 Apothem = a segment that joins the polygon's center to the midpoint of any side that is perpendicular to that side^{[2] X Research source }

2Find the apothem of the polygon. If you're using the apothem method, then the apothem will be provided for you. Let's say you're working with a hexagon that has an apothem with a length of 10√3.Advertisement

3Find the perimeter of the polygon. If the perimeter is provided for you, then you're nearly done, but it's likely that you have a bit more work to do. If the apothem is provided for you and you know that you're working with a regular polygon, then you can use it to find the perimeter. Here's how you do it:
 Think of the apothem as being the "x√3" side of a 306090 triangle. You can think of it this way because the hexagon is made up of six equilateral triangles. The apothem cuts one of them in half, creating a triangle with 306090 degree angles.
 You know that the side across from the 60 degree angle has length = x√3, the side across from the 30 degree angle has length = x, and the side across from the 90 degree angle has length = 2x. If 10√3 represents "x√3," then you can see that x = 10.
 You know that x = half the length of the bottom side of the triangle. Double it to get the full length. The bottom side of the triangle is 20 units long. There are six of these sides to the hexagon, so multiply 20 x 6 to get 120, the perimeter of the hexagon.

4Plug the apothem and the perimeter into the formula. If you're using the formula area = 1/2 x perimeter x apothem, then you can plug in 120 for the perimeter and 10√3 for the apothem. Here is what it will look like:
 area = 1/2 x 120 x 10√3
 area = 60 x 10√3
 area = 600√3

5Simplify your answer. You may need to state your answer in decimal instead of square root form. Just use your calculator to find the closest value for √3 and multiply it by 600. √3 x 600 = 1,039.2. This is your final answer.Advertisement
Part 2
Part 2 of 3:Finding the Area of Regular Polygons Using Other Formulas

1Find 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 the area = 1/2 x 8 x 10, or 40.

2Find the area of a square. To find the area of a square, just square the length of one side. This is really the same thing as multiplying the base of the square by its height, because the base and height are the same.
 If the square has a side length of 6, then the area is 6 x 6, or 36.

3Find the area of a rectangle. To find the area of a rectangle, simply multiply the base times the height.
 If the base of the rectangle is 4 and the height is 3, then the area of the rectangle is 4 x 3, or 12.

4Find the area of a trapezoid. The find the area of a trapezoid, you just have to follow this formula: area = [(base 1 + base 2) x height]/2.
 Let's say you have a trapezoid with bases that have a length of 6 and 8 and a height of 10. The area is simple [(6 + 8) x 10]/2, which can be simplified to (14 x 10)/2, or 140/2, which makes for an area of 70.
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Part 3
Part 3 of 3:Finding the Area of Irregular Polygons

1Write down the coordinates of the vertices^{[3] X Research source } of the irregular polygon. Determining the area for an irregular polygon can be found when you know the coordinates of the vertices.^{[4] X Research source }

2Create an array. List the x and y coordinates of each vertex of the polygon in counterclockwise order. Repeat the coordinates of the first point at the bottom of the list.

3Multiply the x coordinate of each vertex by the y coordinate of the next vertex. Add the results. The added sum of these products is 82.

4Multiply the y coordinate of each vertex by the x coordinate of the next vertex. Again, add these results. The added total of these products is 38.

5Subtract the sum of the second products from the sum of the first products. Subtract 38 from 82 to get 82  (38) = 120.

6Divide this difference by 2 to get the area of the polygon. Just divide 120 by 2 to get 60 and you're all done.Advertisement
Community Q&A

QuestionHow do I calculate the area of an octagon?Community AnswerCut it into smaller shapes, such as triangles or other quadrilaterals using angles. Find the area of each smaller shape, then add the areas together to find the area of the whole shape.

QuestionHow can I calculate the area of an irregular polygon?Community AnswerDivide the polygon into several triangles and/or rectangles. Calculate the area of each, then add them to find the total.

QuestionHow do I find the area of a 4sided shape?DonaganTop AnswererAssuming it's not a square or rectangle, you would have to subdivide the figure into smaller sections consisting of squares, rectangles, triangles, and other shapes whose areas can be easily calculated. Then add those areas together.

QuestionWhat is the area of a pentagon with sides measuring in feet 105, 100, 170, 43, 135?Community AnswerPolygons with more than three sides cannot be determined from side lengths alone. You need to know some of the angles.

QuestionHow can I find the area of a hexagon?Community Answera=1/2ap. a=apothem (like the radius except it comes from the middle of a side). p=perimeter.

QuestionWill this formula work for all regular polygons, regardless of the number of sides?DonaganTop AnswererNot sure which formula you mean. Here's a trig formula that will work for any regular polygon if you know the length of a side: A = s²n / [4 tangent(180°/n)], where s is the length of a side, and n is the number of sides. Here's a trig formula that will work if you know the radius of the polygon (distance from the center to any vertex): A = r²n [sine(360°/n)] / 2, where r is the radius, and n is the number of sides. Finally, here's a trig formula that will work if you know the apothem of the polygon (perpendicular distance from the center to any side): A= a²n[tangent(180°/n)], where a is the length of the apothem, and n is the number of sides.

QuestionAccording to part 3, how do I calculate the area of a circle having a radius of 7 cm, or four points are (0,7), (7,0), (0,7), (7,0)?DonaganTop AnswererPart 3 above has nothing to do with circles. Any circle with a 7 cm radius has an area of 49π or 153.937 square centimeters.

QuestionHow do I calculate the area of a quadrilateral polygon?Community AnswerThere are two ways: Mentally cut the polygon up into other shapes that are easy to calculate, such as triangles and squares. Then calculate each part and sum up the results. Another approach you might take is to imagine the polygon is a shape that can be easily calculated, such as a square. Then find the area of the square. Next, subtract the area you used to fill the square from the square.

QuestionHow do I calculate the area of a polygon?Community AnswerWrite down the formula for finding the area of a regular polygon. To find the area of a regular polygon, all you have to do is follow this simple formula: area = 1/2 x perimeter x apothem. Here is what it means: Perimeter = the sum of the lengths of all the sides.

QuestionHow do I find the area of an eightsided polygon?DonaganTop AnswererThe formula for the area of a regular octagon is 4.828 s², where s is the length of any side. (If the octagon is not regular, no area formula is available.)
Tips
 If you list the points in a clockwise order instead of counterclockwise, you will get the negative of the area. Hence this can be used as a tool to identify the cyclic path or sequence of a given set of points forming a polygon.Thanks!
 This formula computes area with orientation. If you use it on a shape where two of the lines cross like a figure eight, you will get the area surrounded counterclockwise minus the area surrounded clockwise.Thanks!
References
About This Article
To calculate the area of a regular polygon, use the specific formula for that shape. For a triangle, multiply ½ by the length of the base and the height. For a square or rectangle, multiply the length of the base by the height. To find the length of a trapezoid, add the length of the two bases and multiply that number by the height. Then, divide the total by 2 to get the area. Be sure to include units in your answer if they're provided. If you need to calculate the area of an irregularlyshaped polygon, keep reading to learn how!