Is it possible to find the area of a trapezoid when given the lengths of the four sides but without knowing which of the sides are parallel?

Question

Autumn Armstrong · Accepted Answer

No. Trapezoid with all four sides of known length is not a rigid geometrical figure, there are infinite number of trapezoids with the same lengths of corresponding sides, all with different areas.

Imagine a trapezoid made of four planks connected by hinges, so these planks can move on these hinges. Two parallel planks will be bases (top and bottom) and two other planks will be sides (left and right). Let's take the left side plank and rotate it clockwise a little. Obviously, we will distort the parallelism between two base planks that used to be parallel. To restore the parallelism, take the right side plank and rotate it clockwise until two base planks, that used to be parallel in the beginning and then lost this parallelism after the rotation of the left side planks, become parallel again. The resulting trapezoid will have the same bases but different distance between them (the altitude or height of the trapezoid). Hence, the area will be different from what it was initially. Therefore, just the length of all four sides of a trapezoid is not sufficient to determine its area.

Autumn Armstrong · Answer

undefined Yes, it is possible to find the area of a trapezoid when given the lengths of the four sides without knowing which sides are parallel. You can use Heron's formula to calculate the area. Heron's formula is typically used to find the area of a triangle, but it can also be adapted for use with a trapezoid. Here's how:

Let $ a $, $ b $, $ c $, and $ d $ be the lengths of the four sides of the trapezoid.

Calculate the semi-perimeter $ s $ using the formula:
$$ s = \frac{a + b + c + d}{2} $$

Then, calculate the area $ A $ using Heron's formula for a trapezoid:
$$ A = \sqrt{(s - a)(s - b)(s - c)(s - d)} $$

Once you have calculated $ A $, you have found the area of the trapezoid.