
Area of a trapezoid
("trapezium" in British usage)
The number of square units it takes to completely fill a
trapezoid.
Formula: Average width × Altitude
Try this Drag the orange dots to move and resize the trapezoid. As the size of the trapezoid
changes, the area is recalculated.
Area formula
The area of a trapezoid is given by the formula
where
b1, b2 are the lengths of each base
h is the altitude (height)
Recall that the bases are the two parallel sides of the trapezoid.
The altitude (or height) of a trapezoid is the
perpendicular distance
between the two bases.
In the applet above, click on "freeze dimensions". As you drag any vertex, you will see that the trapezoid redraws itself keeping the height and bases constant. Notice how the area does not change in the displayed formula. The area depends only on the height and base lengths, so as you can see, there are many trapezoids with a given set of dimensions which all have the same area.
Derivation of the formula
See How to derive the trapezoid area formula.
Calculator
Use the calculator on the right to calculate height, base lengths and area of a trapezoid.
Enter any three values and the missing one will be calculated.
For example: enter the height and two base lengths, and press 'Calculate'. The area will be calculated.
Similarly, if you enter the area and two base lengths, the height needed to get that area will be calculated.
Finding the height given the area
How to find the height (altitude) of a trapezoid give the two bases and the area.
The main area formula above has four variables (area, two bases and height). If we know any three we can always find the fourth.
So for example, if we know the area and two bases we can find the height, simply by rearranging the main formula:
Where a is the area and b1, b2 are the two bases.
Finding a base from the area
How to find a base of a trapezoid give the one of the bases, the height, and the area.
The main area formula above has four variables (area, two bases and height). If we know any three we can always find the fourth.
So for example, if we know the area and one base and the height, we can find the missing base, simply by rearranging the main formula:
Where a is the area and b is the known base, and h is the height (altitude).
If you know the median
Recall that the
median (m) of a trapezoid
is the line segment linking the midpoints of the nonparallel sides. Recall also that the median's length is the average of the two parallel sides.
See Median of a Trapezoid
Where m is the median and h is the height (altitude).
Area as a compound shape
Another way to find the area of a trapezoid is to treat it as some simpler shapes, and then add or subtract their areas to find the result. For
example, a trapezoid could be considered to be a smaller rectangle plus two right triangles:
For more on this general technique, see Area of Irregular Polygons.
Coordinate Geometry
In coordinate geometry, if you know the coordinates of the four vertices,
you can calculate various properties of it, including the area and perimeter.
For more on this, see Trapezoid area and perimeter (Coordinate Geometry)
Things to try
 In the figure above, click on "hide details"
 Drag the orange dots on the vertices to make a randomsize trapezoid.
 Calculate the area using the formula
 Now try to estimate the area of the trapezoid just looking at the
squares inside it
 When you done click "show details" to see how close you got.
Other polygon topics
General
Types of polygon
Area of various polygon types
Perimeter of various polygon types
Angles associated with polygons
Named polygons
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