How to Calculate the Height of a Trapezoid

The isoceles trapezoid (third from the left on the bottom row) has two bases which are parallel, and two equal sides.

A trapezoid is a four-sided shape that has one set of parallel lines (the bases). If broken down into smaller shapes, it contains 2 right triangles and a rectangle. An isosceles trapezoid has two sides which are the same length, creating two special right triangles, where the other angles are 30 and 60 degrees. Finding the height for the isosceles trapezoid requires a fixed dimension for the side of the trapezoid (which is the hypotenuse of the special right triangle). Finding the height of a non-isosceles trapezoid requires a given side length, as well as the base of the right triangle. For this instruction, assume the side is 6, and the base of the triangle for the second method is 4.

Things You'll Need

• Pencil
• Calculator
• Graph paper
• Ruler

Instructions

Method for an Isosceles Trapezoid

1. 

   This glasswork is made up of special right triangles.

   Using your ruler, draw a straight line from the top left side of the trapezoid to the point on the bottom directly below it. This gives the first special right triangle.

2. The shorter leg, or the portion remaining on the longer base, is one half the distance of the hypotenuse, or the side of the trapezoid. If the side of the trapezoid is 6, then the shorter leg is 3.

3. The longer leg of the right triangle -- in this case the trapezoid height -- is the length of the shorter leg multiplied by the square root of three. Given the shorter leg distance of 3, multiply that distance by the square root of three. This will most likely require use of the calculator. The resulting solution is the height of the isosceles trapezoid. Using the other dimensions of 6 and 3, the answer is 5.2 (rounded up to one decimal place).

Method for Any Trapezoid (Using the Pythagorean Theorem)

1. As in Step 1 above, draw a line from the top corner of the trapezoid to its matching point on the base below. This creates a right triangle.

2. Using the side length of the trapezoid, solve for the hypotenuse. The Pythagorean Theorem gives the sides of a right triangle as a^2 + b^2 = c^2, where c is the hypotenuse. Given the side of the trapezoid as a distance of 6, and 6 times itself (squared) is 36, this means that the hypotenuse of the new right triangle squared is 36.

3. Square the base. Given that the base is 4, this plugs into the equation as 16.

4. If a^2 + b^2 = c^2, then a^2 + 16 = 36. Solve for "a" by subtracting 16 from 36; and find that the height of the trapezoid is the square root of 20 (4.47214, rounded to the nearest decimal place).

References

• Math Open Reference
• University of Georgia
• CalcEinstein

Resources

• Read this Article in Spanish