The sides of the base of the rectangular parallelepiped are 6 and 8 cm and its diagonal is inclined to the base plane at an angle of 45 degrees. Find: 1. the area of the lateral surface of the parallelepiped 2. the total surface area of the parallelepiped.
At the base of a rectangular parallelepiped is a rectangle. Its diagonal is a right-angled triangle with two adjacent sides. Let us find the length of the diagonal of the base by the Pythagorean theorem:
d = √ (8² + 6²) = √ (64 + 36) = √100 = 10 (cm).
The diagonal of the box, the diagonal of the base, and the height of the box make up a rectangular triangle (the height is perpendicular to the base). One of the angles is 45 °, the second 90 °, which means that the third angle is also 45 °.
Hence, this is a right-angled triangle with equal legs. Therefore, the height of the parallelepiped is 10 (cm).
1) The side surface consists of four rectangles (which are equal in pairs), we find their area:
8 * 10 = 80 (cm²).
6 * 10 = 60 (cm²).
Hence, the lateral surface area is equal to:
Side = 80 * 2 + 60 * 2 = 160 + 120 = 280 (cm²).
2) The total surface area is the sum of the lateral surface area and the base area.
Sb = 6 * 8 = 48 (cm²).
Sp.p = Sbok + 2 * Sbn = 280 + 48 * 2 = 280 + 96 = 376 (cm²).
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