In an isosceles trapezoid, the height and base are 3: 5: 13, and the side is 15 cm. find the area of the trapezoid.

Let’s replace the ratio of the height, and the bases through x.
Height H = 3 * x; base b = 5 * x; base a = 13 * x. Side c = 15 cm.

The area of the trapezoid is determined by the formula:
S = (a + b) * H / 2 = (5 + 13) * x * 3 * x / 2 = 27 * x ^ 2.
It is necessary to determine a, b, and H or x.

The height H is the lateral side c, and half of the difference of the bases make up a right-angled triangle, in which the Pythagorean theorem is valid:
c ^ 2 = H ^ 2 + [(a – b) / 2] ^ 2;
15 ^ 2 = (3 * x) ^ 2 + [(13 * x – 5 * x)] / 2 ^ 2;

225 = 9 * x ^ 2 + (4 * x) ^ 2; 225 = (9 + 16) * x ^ 2; x ^ 2 = 9;
x = √ (9) = 3.

We define S = (a + b) * H / 2 = 27 * x ^ 2 = 27 * 9 = 243 (see square).