The diagonal of a rectangular parallelepiped is √194 dm, and the diagonals of its side faces are √130

| April 6, 2021 | education

The diagonal of a rectangular parallelepiped is √194 dm, and the diagonals of its side faces are √130 dm and √ 145 dm. Define the complete surface of the parallelepiped.

Consider a triangle DА1В1. Since A1B1 is perpendicular to the plane АА1DD1, then the diagonal А1D is perpendicular to А1D1, and therefore, the triangle DА1D1 is rectangular. Then, by the Pythagorean theorem, A1D1 ^ 2 = DD1 ^ 2 – DA1 ^ 2 = (√194) ^ 2 – (√145) ^ 2 = 194 – 145 = 49.

A1D1 = √49 = 7 cm.

Since ABDCA1B1C1D1 is a parallelepiped, then A1D1 = AB = CD = C1D1 = 7 cm.

Consider a right-angled triangle DСС1, and by the Pythagorean theorem define the leg CC1.

CC1 ^ 2 = DC1 ^ 2 – CD ^ 2 = (√130) ^ 2 – 72 = 130 – 49 = 81.

CC1 = √181 = 9 cm.

Since ABDСА1В1С1D1 is a parallelepiped, then АА1 = BB1 = CC1 = DD1 = 9 cm.

Consider a right-angled triangle ADA1 and, by the Pythagorean theorem, define the leg AD.

AD ^ 2 = DA1 ^ 2 – AA1 ^ 2 = (√145) ^ 2 – 9 ^ 2 = 145 – 81 = 64.

AD = √64 = 8 cm.

Since ABDСА1В1С1D1 is a parallelepiped, then АD = ВС = А1D1 = В1С1 = 8 cm.

Determine the total surface area of ​​the parallelepiped.

Spol = 2 * AD * AA1 + 2 * AB * AA1 + 2 * AB * BC = 2 * 8 * 9 + 2 * 7 * 9 + 2 * 7 * 8 = 144 + 126 + 112 = 382 cm2.

Answer: The total surface area is 382 cm2.



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