The sides of the base of the rectangular parallelepiped are 4cm and 6cm, and its diagonal makes an angle of 60

The sides of the base of the rectangular parallelepiped are 4cm and 6cm, and its diagonal makes an angle of 60 * with the base plane. Find the total surface area of the parallelepiped.

Let us introduce the notation: a and b are the sides of the base of this rectangular parallelepiped, d is the diagonal of the base, D is the diagonal of the parallelepiped, h is its height.

At the base of a rectangular parallelepiped is a rectangle, its area is defined as the product of the sides:

Sb = a * b = 4 * 6 = 24 cm2.

We find the diagonal of the base d by the Pythagorean theorem:

d2 = a2 + b2 = 42 + 62 = 16 + 36 = 52;

d = √52 = 2√13 cm.

The diagonal of the parallelepiped D, the diagonal of the base d and the side edge equal to the height h form a right-angled triangle.

By condition, the angle between the diagonal of the parallelepiped and the plane of the base is 60 °.

The ratio of the opposite leg to the adjacent leg is equal to the tangent of the angle, which means:

h / d = tg 60 °;

h = d * tg 60 ° = 2√13 * √3 = 2√39 cm.

The area of ​​the lateral surface of the parallelepiped is equal to the product of the height and the perimeter of the base:

Sside = Rosn * h = 2 * (4 + 6) * 2√39 = 40√39 cm2.

The total surface area is equal to the sum of the lateral surface areas and the two bases:

S full = S side + 2 * Sb = 40√39 + 2 * 24 = 40√39 + 48 ≈ 297.8 cm2.