A *trapezoid* is a [quadrilateral](http://www.ehow.com/how_5244404_learn-quadrilateral-shapes.html) -- a four sided shape -- with two parallel sides, called *bases*, of unequal length. You can find the area of a trapezoid using the lengths of these two sides, and the distance between the two sides, called the **height**. If you don't know a trapezoid's height, solve for it using the [Pythagorean Theorem](http://www.ehow.com/video_4754335_what-pythagorean-theorem.html) and length of the trapezoid's **diagonals**, the two sides that connect its bases.
Area Formula for a Trapezoid
A trapezoid's area depends on the length of its two parallel bases and the perpendicular distance between those two bases. Area, A, is given by the lengths of the two bases, b1 and b2, divided by 2 and multiplied by the trapezoid's height, h, as follows:
A = (b1 + b2) / 2 x h
Take a trapezoid whose bases are 4 and 6 inches long and which is 2 inches high. Add the base lengths, 4 and 6, and divide by 2 to get 5 -- 5 multiplied by the height, 2, is 10, so the trapezoid has an area of 10 square inches.
Height and Triangles
If you don't know the height of a trapezoid, solve for it by imagining height as one side of a right triangle and using the Pythagorean Theorem. This triangle will have the length of one of the trapezoid's diagonals as its hypotenuse, the height as one leg and a section of the base as the other leg. Unless you know this length, which is not the length of the base, you cannot use the Pythagorean Theorem. The Pythagorean Theorem relates the length of the diagonal, c, to height as follows:
c^2 = h^2 + l^2
Here h is the height and l is the length of the section on the trapezoid's base. The ^2, represents squaring. The exponent sign ^ signifies that you are multiplying the number by itself, and the exponent 2 signifies the instances of that number multiplied by itself. So c^2 = c x c.
Solving Using the Pythagorean Theorem
You can now solve for height by subtracting the base section length squared, l^2, from the diagonal length squared, c^2. Take a trapezoid with a diagonal of 5 inches that meets the shorter base 4 inches horizontally from where it meets the longer base. Find the length of height squared by subtracting 4 squared from 5 squared: h^2 = 5^2 - 4^2 = 25 - 16 = 9. So the height squared is 9, meaning the height is equal to the square root of 9, which is 3. If the base lengths of this trapezoid are 10 inches and 18 inches, you can now solve for area: A = (10 + 18) / 2 x 3 = 14 x 3 = 42. So the trapezoid has an area of 42 square inches.
Note that if one of the diagonals intersects with both bases at a right angle, the leg that is part of the trapezoid's base has a length of zero, and the length of the diagonal is equal to the height.
Height of a Isosceles Trapezoid
An isosceles trapezoid has two diagonals of equal length. Because of this, unlike other trapezoids, you can always calculate the length of the base section, l, covered by each diagonal of an isosceles trapezoid. In this case, the section is the length of the larger base minus the length of the smaller base, divided by two: l = (b1 - b2) / 2
Take an isosceles trapezoid with bases of 12 and 24 inches long, and diagonals 10 inches long. Solve for height by first finding the section length: l = (24 - 12) / 2 = 6 and plugging it into the Pythagorean Theorem h^2 = 10^2 - 6^2 = 100 - 36 = 64. Height is equal to the square root of 64, which is 8.
You now have both base lengths and the height, and so can solve for area: A = (12 + 24) / 2 x 8 = 18 x 8 = 144 So this trapezoid has an area of 144 square inches.