A regular cube or hexahedron is a volumetric geometric figure, a solid body that has six equal square-shaped faces. It is a right rectangular parallelepiped, and it is also a right rectangular prism with the height and the sides of the base of equal length. In a simpler and more familiar way, a cube can be thought of as a cardboard box made up of six squares of equal size. Let’s see how you can determine the area of a cube.
The formula to determine the area or volume of a right prism implies knowing the length of the sides of the base and the height, which in the general definition of a rectangular prism are different. But in the case of a cube the formula is simplified by being equal to the three lengths. Anyway , let’s first see how to calculate the area of a right rectangular prism.
A prism is a polyhedron, a solid body formed by flat faces, which has two equal and parallel faces called bases, while the lateral faces are parallelograms, four-sided flat figures whose opposite sides are equal and parallel. A triangular prism is one that has a triangle as its base, while a rectangular or quadrangular prism is one that has a rectangle as its base, a pentagonal prism has a pentagon as its base, and so on. A right prism is one in which the lines joining the lateral faces as well as the planes containing them are perpendicular to the bases. The following figure shows right prisms with different bases.
A right rectangular prism has rectangles for bases and side faces, as shown in the figure below. Thus, the area of a right rectangular prism will be the sum of the area of the four rectangles that form the lateral faces added to the area of the rectangles that form the bases.
If the bases are rectangles with width a and length l , as shown in the figure, the area of each of these rectangles will be a × l . The lateral faces are rectangles whose sides are h and a on two faces, and h and l on the other two. The areas of these rectangles will be a × h and l × h . Adding the area of the six rectangles gives the area A p of the right rectangular prism.
A p = 2 × a × l + 2 × a × h + 2 × l × h
The volume V p of a right rectangular prism is calculated as:
V p = a × l × h
If we now have a cube that, as said, is a right rectangular prime with the sides of the base and the height of equal length c , c = a = l = h , the area A c of a cube of side c will be:
A c = 6 × c × c or A c = 6 × c 2
And the volume V c of a cube of side c will be
V c = c × c × c or V c = c 3
In the specific case of a cube that has a side of 5 centimeters, we can calculate the area by substituting the value 5 in the previous formula for A c and we will obtain
A c = 6 × 5 × 5
A c = 150
The area of a cube with a side of 5 centimeters is 150 square centimeters (150 cm 2 ).
In the same way, to calculate the volume of this cube we substitute the value 5 in the formula of V c , and we obtain
V c = 5 × 5 × 5
V c = 125
The volume of a cube with a side of 5 centimeters is 125 cubic centimeters (125 cm 3 ) .
Fountain
Alexei V Pogorelov. Elemental geometry . Mir Publishing House, Moscow.
