To find the area of an isosceles trapezoid, we need to know the lengths of the parallel sides and the height (or altitude) of the trapezoid.
In this case, we know that one parallel side is 20 centimeters long and the other parallel side is 32 centimeters long. However, we don't know the height of the trapezoid.
To find the height of the trapezoid, we can draw a line perpendicular to the parallel sides, creating two right triangles.
The height of the trapezoid is the hypotenuse of one of these right triangles, and we can use the Pythagorean theorem to find its length.
The legs of the right triangle are:
- Half of the difference between the parallel sides: (32 - 20) / 2 = 6
- The height of the trapezoid (which we'll call h)
Using the Pythagorean theorem, we can write:
h^2 = 6^2 + x^2
where x is the length of the height of the trapezoid.
Simplifying, we get:
h^2 = 36 + x^2
We still don't know the value of x, but we do know that the height of the trapezoid is perpendicular to the bases, so it forms a rectangle with the shorter base. Therefore, the height is also the length of the two sides of a right triangle with a hypotenuse of 20 (half of the shorter base).
Using the Pythagorean theorem again, we can write:
h^2 + 6^2 = 20^2
Simplifying, we get:
h^2 = 400 - 36
h^2 = 364
h ≈ 19.06
Now that we know the height of the trapezoid, we can use the formula for the area of a trapezoid:
Area = (base1 + base2) / 2 x height
Plugging in the values we know, we get:
Area = (20 + 32) / 2 x 19.06
Area ≈ 526.24 square centimeters
Therefore, the area of the isosceles trapezoid is approximately 526.24 square centimeters.