Using the area, the height is found by dividing twice the area by the sum of the two bases of the trapezoid. Without the area, the height is found by using the Pythagorean Theorem; right triangles can be formed from the ends of the trapezoid. The slanted edges form the hypotenuses, and the length of the triangle's bottom sides is half of the difference between the two bases.
The Pythagorean Theorem states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the legs. This equation can be modified to find one of the legs if the hypotenuse and the other leg is known; in this case, the tall leg is the height of the trapezoid. The legs of a right triangle are defined as the sides of the triangle opposite from the acute angle and form the shorter two sides of the triangle. The equation that uses the area of the trapezoid is found by manipulating the original equation for the area of the trapezoid and solving it for the height. Trapezoids have plenty of uses outside of math. In architecture, trapezoids are used to refer to symmetrical doors, windows or buildings that are built wider at the base and tapered towards the top. This style was the standard style of architecture used by the Incas.