Surface Area of Prisms and Geometric Prisms and Cube and Rectangular Prisms

Definition and Fundamentals of Surface Area

• Surface area ($SA$) is defined as the total sum of all the areas of its surfaces on a three-dimensional object.

• A Net is a two-dimensional representation or a diagram that shows all the surfaces of a three-dimensional figure when unfolded or flattened out.

Geometric Properties of a Cube

• A cube is a special type of prism where all faces are equal squares.

• The area ($A$) of a single face of a cube can be calculated using the length of its side ($s$) or length ($l$):

  • $A = s^2$

  • $A = l^2$

• The total Surface Area ($SA$) for a cube, consisting of six identical faces, is given by the formula:

  • $SA = 6 \times s^2$

Rectangular Prisms: Structure and Calculations

• A rectangular prism consists of several faces, typically defined as:

  • Top

  • Back

  • Front

  • Sides

• The dimensions used to describe a rectangular prism include length ($l$), base ($b$), and height ($h$).

• The area of a face can be calculated as:

  • $A = lw$

  • $A = lb$

• The total Surface Area ($SA$) of a rectangular prism is calculated using the following formula:

  • $SA = 2(b + h + bh)$

Triangular Prisms: Surface Area Components

• Triangular prisms are identified by their triangular bases and rectangular sides, labeled in certain contexts as components A, B (referenced as 6), and C.

• The total Surface Area ($SA$) of a triangular prism is the sum of the areas of its two triangular bases and its three rectangular faces:

  • $SA = 2(\frac{1}{2}bh) + A(A) + A(6) + A(C)$

• Factors for specific side areas, such as side A, are calculated by multiplying the length and the width of that specific rectangular face:

  • $A(A) = lw$