The bases of an isosceles trapezoid circumscribed about a circle are 54 and 24. What is the height of the trapezoid?

The radius of a circle inscribed in an isosceles trapezoid can be found using two formulas:
1.r = √ (a * b) / 2,
where r is the radius of the inscribed circle, a and b are the smaller and larger of the base of the isosceles trapezoid, respectively.
2.r = h / 2,
where h is the height of the isosceles trapezoid.
Thus, you can compose equality:
h / 2 = √ (a * b) / 2.
We substitute the data we know, solve the resulting equation and find the height of the trapezoid:
h / 2 = √ (24 * 54) / 2;
h / 2 = √1296 / 2;
h / 2 = 36/2;
h / 2 = 18;
h = 18 * 2 = 36 (cm).
Answer: h = 36 cm.