The diagonals of the bases of a regular rectangular truncated pyramid are equal to 3√2 and 9 √2

The diagonals of the bases of a regular rectangular truncated pyramid are equal to 3√2 and 9 √2, and the side edge is inclined to the base plane at an angle of 60 degrees. Find S of the diagonal section of the pyramid

The diagonal section of a regular rectangular truncated pyramid is an isosceles trapezoid, the bases of which D and d are diagonals of the bases of the truncated pyramid.

The projection of the lateral side of such a trapezoid onto the larger base is equal to half the difference in the lengths of the bases of the trapezoid:

a = (D – d) / 2 = (9√2 – 3√2) / 2 = 6√2 / 2 = 3√2.

The projection of the side of the trapezoid onto the larger base, the side and height form a right-angled triangle. The ratio of the opposite leg to the adjacent leg is equal to the tangent of the angle, which means:

tg 60 ° = h / a;

h = a * tg 60 ° = 3√2 * √3 = 3√6 – trapezoid height.

The area of ​​the trapezoid is equal to the product of the length of the midline by the height:

S = h * m = h * (D + d) / 2 = 3√6 * (9√2 + 3√2) / 2 = 3√6 * 12√2 / 2 = 3 * 6 * √2 * √ 6;

S = 36√3 cm2 – the area of ​​the diagonal section of this pyramid.