Two edges of a rectangular parallelepiped extending from one vertex are equal to 48 and 12 diagonal
Two edges of a rectangular parallelepiped extending from one vertex are equal to 48 and 12 diagonal of a parallelepiped is equal to 52 find the surface area and volume of the parallelepiped.
We will use the theorem on the diagonal of a rectangular parallelepiped: the square of the diagonal of a rectangular parallelepiped is equal to the sum of the squares of its three dimensions.
In the drawing: a – length, b – width, c – height, d – diagonal.
d ^ 2 = a ^ 2 + b ^ 2 + c ^ 2.
42 ^ 2 + 12 ^ 2 + c ^ 2 = 52 ^ 2.
2 304 + 144 + c ^ 2 = 2 704.
2,448 + c ^ 2 = 2,704.
c ^ 2 = 2 704 – 2 448.
c ^ 2 = 256.
c = √256.
c1 = 16; c2 = -16 (the second root is not suitable, because c is the height of the box, the value of which cannot be expressed as a negative number).
Find the surface area of the parallelepiped. It has 6 faces, each face is a rectangle. You need to find the areas of each face and add them. The formula can be written like this:
S on top. = 2ac + 2ab + 2bc = 2 x (ac + ab + bc).
S on top. = 2 x (48 x 16 + 48 x 12 + 12 x 16) = 2 x (768 + 576 + 192) = 2 x 1 536 = 3 072.
We find the volume of the parallelepiped by the formula: V = a x b x c.
V = 48 x 12 x 16 = 9 216.
Answer: the surface area of the parallelepiped is 3,072, its volume is 9,216.