Rectangular Solids

A rectangular solid or box is a solid formed by six rectangles, called faces. The sides of the rectangles are called edges. As in the diagram below, the edges are called the length, width, and height. A cube is a rectangular solid in which the length, width, and height are equal, so all the edges are the same length.

Rectangular solid with dotted lines representing the edges, h = height, l = length, w = width, and a grayed out side representing the face. Below is a cube with edges = e and a grayed out side for the face.

The volume of a solid is measured in cubic units. One cubic unit is the amount of space taken up by a cube all of whose edges are one unit long. In the figure on page 171, if each edge of the cube is 1 inch, the area of each face is 1 square inch, and the volume of the cube is 1 cubic inch.

Key Fact K1

The surface area of a rectangular solid is the sum of the areas of the six rectangular faces. The areas of the top and bottom faces are equal, the areas of the front and back faces are equal, and the areas of the left and right faces are equal. Therefore, to get the total surface area, we can calculate the area of one face from each pair, add them up, and then double the sum. In a cube, each of the six faces has the same area, so the surface area is six times the area of any face.

Rectangular prism where each face is labeled with A = lw. Below is a cube where each face is labeled A = e^2.

Key Fact K2

A diagonal of a rectangular solid is a line segment joining a vertex on one face of the box to the vertex on the opposite face that is furthest away. A rectangular solid has four diagonals, all the same length. In the following box, diagonals AG¯ and BH¯ are drawn in. The other two diagonals are CE¯ and DF¯.

Rectangular prism with vertices at ABCDEFGH. Dotted lines from opposite vertices indicate diagonals AG and BH.

Key Fact K3

If the dimensions of a rectangular solid are , w, and h and if d is the length of a diagonal, then d=2+w2+h2.

The formula given in KEY FACT K3 is obtained by using the Pythagorean theorem twice. In the figure below, BD¯ is the diagonal of rectangular face BCDE. By the Pythagorean theorem, (BD)2 = 2 + w2. Now ΔADB is a right triangle, and by the Pythagorean theorem:

d2=(AB)2=(BD)2+(AD)2=(2+w2)+h2d=2+w2+h2
Rectangular prism with face BCDE. Diagonal in rectangle face is BD. Side BC is the length, side CD is the width, and side AD is the height. Rectangular prism diagonal is AB with a length of d.