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.

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 from 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 re-arranging 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 re-arranging 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 non-parallel sides. Recall also that the median's length is the average of the two parallel sides.
See Median of a Trapezoid

If the median length is m, and the altitude h, the area of the trapezoid is

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 random-size 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.