Surface Areas and Volumes - Surface Area of a Cuboid and a Cube

A cuboid is a three-dimensional solid figure bounded by six rectangular faces. Visually, it resembles a rectangular box where opposite faces are identical in shape and size and are parallel to each other. It is defined by three dimensions: length ($l$), breadth ($b$), and height ($h$).

A cube is a special case of a cuboid where all three dimensions are equal (length = breadth = height = $a$). Visually, it consists of six identical square faces, much like a standard playing die or a Rubik's cube, making it perfectly symmetrical.

The Total Surface Area (TSA) of a cuboid is the sum of the areas of all its six rectangular faces. If you were to flatten a cuboid into a 'net' diagram, you would see three pairs of congruent rectangles: two of size $l \\times b$ (top and bottom), two of size $b \\times h$ (side faces), and two of size $h \\times l$ (front and back).

The Lateral Surface Area (LSA) of a cuboid refers to the sum of the areas of only the four vertical faces, excluding the top and bottom. Visually, this is equivalent to the area of the four walls of a rectangular room.

For a cube, the Total Surface Area (TSA) is simpler to calculate because all six faces are identical squares, each having an area of $a^2$. Therefore, the total area is simply $6$ times the area of one face.

The Lateral Surface Area (LSA) of a cube is the area of its four vertical square faces. Visually, if you look at a cube from the sides without looking at the top or the base, those four square faces constitute the LSA.

The diagonal of a cuboid represents the longest straight line segment that can fit inside the box. Visually, this line connects one bottom corner to the opposite top corner, passing through the interior center of the cuboid.

Surface area is a measure of the two-dimensional boundary of a three-dimensional object. Consequently, the units are always squared, such as $cm^2$ or $m^2$, as they are derived from the product of two linear dimensions.