Perimeter of a trapezoid

The perimeter of a trapezoid where a, b, c, d are the lengths of its sides can be found using the formula p = a + b + c + d.

Example #1: Find the perimeter for the trapezoid below

Perimeter  = 5 + 6 + 7 + 10 = 11 + 17 = 28

Now, let us make things a little bit more interesting. How about the figure below?

Example #2: Can you find the perimeter for this right trapezoid where ABC is a right triangle?

This example is a little tricky! Although 4 sides are given, one of them cannot be used to find the perimeter of the trapezoid. Can you tell which one? Yes, you guessed it right! The side that you cannot use is AC  = 10.

The sides that you can use are AB, AD, BC, and CD. However, we have another problem. BC is missing. We need to find BC.

BC is one side of the right triangle ABC. Therefore, we can use the Pythagorean theorem to find BC. 

AC² = AB² + BC²

10² = 8² + BC²

100 = 64 + BC²

100 - 64 = BC²

36 = BC²

Since 6² = 36, BC = 6

Perimeter = AB + BC + CD + AD  = 8 + 6 + 10 + 12 = 14 + 22 = 36

Finding the perimeter of a trapezoid when the height, the length of the top base, and the lengths of the nonparallel sides are given.

Example #3: Find the perimeter of the following trapezoid where the length of the bottom base is not known.

Make a rectangle by drawing a line from the other vertex on top. 

Now, once we find the length of x and the length of y, we can find the perimeter.

To find the length of y, we just need to use the Pythagorean theorem

y² + 8² = 10²

y² + 64 = 100

y²  = 100 - 64

y²  = 36

Since 6² = 36, y = 6

Find x the same way.

x² + 8² = 9²

x² + 64 = 81

x²  = 81 - 64

x²  = 17

x  = √17 = 4.12

Perimeter = 6 + 7 + 4.12 + 9 + 7 + 10 = 43.12

Finding the perimeter of a trapezoid when the height, the length of the top base, and the bottom base angles are known.

Example #4: Find the perimeter of the following trapezoid where the length of the bottom base and the lengths of the nonparallel sides are not known.

To find a, b, x, and y, we can use trigonometric ratios.

sin(60 degrees) = 15/a or a  = 15/sin(60 degrees) = 15/0.866 = 17.3

tan(60 degrees) = 15/y or y  = 15/tan(60 degrees) = 15/1.732 = 8.66

sin(70 degrees) = 15/b or b  = 15/sin(70 degrees) = 15/0.94 = 15.95

tan(70 degrees) = 15/x or x  = 15/tan(70 degrees) = 15/2.74 = 5.47

Perimeter = 17.3 + 12 + 15.95 + 5.47 + 12 + 8.66 = 71.38