# Find the area of the lateral surface of a rectangular parallelepiped, the sides of the base of which are 12cm

Find the area of the lateral surface of a rectangular parallelepiped, the sides of the base of which are 12cm and 16cm, and the diagonal of the parallelepiped makes an angle of 45 degrees with the plane of the base.

The diagonal of a parallelepiped is a hypotenuse in a right-angled triangle, in which the diagonal of the base and the side edge are legs. Since the diagonal of the parallelepiped is inclined to the plane of the base at an angle of 45 °, this triangle is isosceles, which means that the side edge is equal to the diagonal of the base.

The square of the diagonal of the base can be found as the sum of the squares of the sides of the base:

d ^ 2 = 12 ^ 2 + 16 ^ 2 = 144 + 256 = 400 = 20 ^ 2;

d = 20 cm – diagonal of the base.

h = d = 20 cm – lateral rib.

The lateral surface area of ​​a rectangular parallelepiped is equal to the product of the base perimeter by the length of the lateral rib:

Sside = Rosn * h = (16 * 2 + 12 * 2) * 20 = 56 * 20 = 1120 cm2.

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