# In a rectangular parallelepiped, the sides of the base are 10 cm and 17 cm; one of the base diagonals is 21 cm;

**In a rectangular parallelepiped, the sides of the base are 10 cm and 17 cm; one of the base diagonals is 21 cm; the largest diagonal of the parallelepiped is 29 cm. Determine the full surface of the parallelepiped.**

Let us determine the area of the AСD triangle at the base of the AВСD by Heron’s theorem.

The half-perimeter of the triangle is: p = (17 + 21 + 10) / 2 = 48/2 = 24 cm.

Then Sacd = 24 * (24 – 21) * (24 – 17) * (24 – 10) = 24 * 3 * 7 * 14 = 7056 = 84 cm2.

Then Sbn = 2 * Sacd = 2 * 84 = 168 cm2.

From the right-angled triangle ACC1, according to the Pythagorean theorem, we determine the leg CC1.

CC12 = AC12 – AC2 = 841 – 441 = 400.

CC1 = 20 cm.

Let’s define the lateral surface area. Side = Ravsd * CC1 = 54 * 20 = 1080 cm2.

Then Sпов = 2 * Sсн + S side = 336 + 1080 = 1416 cm2.

Answer: The surface area is 1416 cm2.