Trapezoid area and perimeter(Coordinate Geometry)
 
The area and perimeter of a trapezoid can be calculated if you know the
coordinates of its vertices.
Try this Drag any vertex of the trapezoid below. It will remain a trapezoid and the area and perimeter calculated. You can also drag the origin point at (0,0), or move the rectangle itself.

Area

The area of a trapezoid is calculated by multiplying the average width by the altitude. See Trapezoid definition (coordinate geometry) to see how the side lengths and altitude are found. (Note too that the median length is the same as the average width.)

As a formula:


where
b1, b2 are the lengths of the two bases (BC and AD)
a is the altitude of the trapezoid

In the figure above, drag any vertex of the trapezoid and note how the area is calculated.

Perimeter

The perimeter of a trapezoid is simply the sum of all four sides. Since they have no relationship to each other, there is no formula for it. Simply find the four side lengths and add them up. The length of each side is found using the techniques described in Distance between two points (given their coordinates) which are used to find the distance between each side's endpoints.

Example

Find the area and perimeter of the trapezoid in the figure above. First press 'reset' and 'show altitude'.

Area

  1. First, we need the length of the two bases (the parallel sides). These are found by calculating the distance between the endpoints of the lines segments. (See Distance between two points). Doing this we see that
    BC = 22 and AD = 47
  2. Then we need the altitude. This is the perpendicular distance between the bases. As described in Trapezoid (coordinate geometry) there are several ways to do this depending on whether the trapezoid is rotated or not. Doing this we see that
    altitude = 21
  3. Finally we calculate the area as the altitude times the average width ( average base length):
    Equation Calculator
    Which agrees with the calculated figure above.

Perimeter

  1. The perimeter is the sum of the four side lengths. So these are found by calculating the distance between the endpoints of the lines segments. (See Distance between two points). Doing this we see that
    BC = 22   AD = 47   AB=22   CD=28
  2. Finally we add them up to get the perimeter
    22 + 22 + 28 + 47 = 119
    Which agrees with the calculated figure above.

Rotated case

In the figure above the trapezoid bases are parallel to the x-axis which makes calculations easy. If you click 'rotated' this will not be the case. All the techniques described above will still work, but you have to use the correct method for finding the distance between two points, and the altitude, which requires the correct method for finding the perpendicular distance from a point to a line. See

Things to try

  1. Click on "hide details" and "rotated" then drag the vertices of the trapezoid around to create an arbitrary size.
  2. From the coordinates of the corner points, calculate the area and perimeter of the trapezoid.
  3. Then click on "show details" to check your result. (The results shown above are rounded off to whole numbers for clarity)

Limitations

In the interest of clarity in the applet above, the coordinates are rounded off to integers and the lengths rounded to one decimal place. This can cause calculatioons to be slightly off.

For more see Teaching Notes

Other Coordinate Geometry entries