Surface Area of Rectangular and Triangular Prisms - Sixth Grade Mathematics

The surface area of a prism is the total area of all its two-dimensional outer faces combined, measured in square units.

The Surface Area of Rectangular Prisms

A rectangular prism has six rectangular faces consisting of three congruent opposite pairs: front and back, top and bottom, left and right.

Rectangular Prism Dimensions:
                       +----------------+
                      /                /|
                     /                / | Height h
                    +----------------+  |
                    |                |  +
                    |                | / Depth/Width w
                    |                |/
                    +----------------+
                        Length l

Faces and Formulas:
- Top and Bottom faces:     Area = 2 * (l * w)
- Front and Back faces:     Area = 2 * (l * h)
- Left and Right faces:     Area = 2 * (w * h)

Total Surface Area (SA) Formula:
SA = 2lw + 2lh + 2wh   or   SA = 2(lw + lh + wh)

Step-by-Step Example:
Find the surface area of a rectangular cardboard box with length 8 cm, width 5 cm, and height 4 cm.
1. Top and Bottom:   2 * (8 * 5) = 2 * 40 = 80 cm^2
2. Front and Back:   2 * (8 * 4) = 2 * 32 = 64 cm^2
3. Left and Right:   2 * (5 * 4) = 2 * 20 = 40 cm^2
Total Surface Area = 80 + 64 + 40 = 184 square centimeters.

For a cube where all edges are equal to length s: SA = 6 * s^2. For example, a cube with edge 5 inches has SA = 6 * 5^2 = 6 * 25 = 150 square inches.

The Surface Area of Triangular Prisms

A triangular prism has five total faces: two congruent triangular bases and three rectangular lateral faces.

Anatomy of a Triangular Prism:
                      /\
                     /  \  Triangular Base (Base b, Height h_t)
                    /    \
                   +------+----------------+
                   |     /                /|
                   |    /                / | Length of Prism L
                   |   /                /  |
                   |  /                /   +
                   | /                /   /
                   |/                /  /
                   +----------------+--+

Step-by-Step Triangular Prism Protocol:
Face 1 & 2 (Two Triangular Bases):
   Area of one triangle = (1 / 2) * b * h_t
   Area of BOTH triangles = 2 * [(1 / 2) * b * h_t] = b * h_t

Faces 3, 4, & 5 (Three Rectangular Lateral Sides):
   Rectangle 1 = Side 1 * Length of Prism
   Rectangle 2 = Side 2 * Length of Prism
   Rectangle 3 = Side 3 * Length of Prism

Total Surface Area = (2 * Area of Base) + (Perimeter of Base * Prism Length)
SA = 2B + P * L

Worked Example:
A right-triangular prism has triangular base legs of 6 cm and 8 cm with hypotenuse 10 cm.
The prism length is 15 cm.
Step 1: Triangular bases (2):
        Base Area = (1 / 2) * 6 * 8 = 24 cm^2.
        Both bases = 2 * 24 = 48 cm^2.
Step 2: Rectangular sides (3):
        Face 1 = 6 * 15 = 90 cm^2.
        Face 2 = 8 * 15 = 120 cm^2.
        Face 3 = 10 * 15 = 150 cm^2.
        Sum of lateral faces = 90 + 120 + 150 = 360 cm^2.
Step 3: Total Surface Area = 48 + 360 = 408 square centimeters.

Real-World Applications: Packaging and Painting

Surface area problems frequently involve practical constraints where certain faces are omitted. For example, painting the four walls of a room excludes the floor and ceiling (only 2lh + 2wh). Wrapping a gift or coating an object with paint requires calculating full external surface area.

Exercises

Exercise 1. Calculate the total surface area of each rectangular prism: (a) Length 10 cm, width 6 cm, height 4 cm; (b) A cube with side length 7 inches; (c) Length 12.5 meters, width 4 meters, height 2 meters.

Exercise 2. A triangular prism has a right-triangle base with sides 5 cm, 12 cm, and 13 cm. The length of the prism is 20 cm. Find the total surface area.

Exercise 3. A cereal box manufacturer wants to determine the amount of cardboard required to manufacture a box measuring 12 inches high, 8 inches wide, and 2.5 inches deep. Calculate the total surface area of the box.

Exercise 4. A wooden toy chest with an open top (no lid) has length 4 feet, width 2.5 feet, and height 2 feet. Find the total exterior surface area that needs to be painted.

Solutions

Solution to Exercise 1. For (a), SA = 2(10 * 6 + 10 * 4 + 6 * 4) = 2(60 + 40 + 24) = 2(124) = 248 cm^2. For (b), SA = 6 * s^2 = 6 * 7^2 = 6 * 49 = 294 in.^2. For (c), SA = 2(12.5 * 4 + 12.5 * 2 + 4 * 2) = 2(50 + 25 + 8) = 2(83) = 166 m^2.

Solution to Exercise 2. The two right-triangular bases have legs 5 cm and 12 cm: Area of one base = (1/2) * 5 * 12 = 30 cm^2. Both bases = 2 * 30 = 60 cm^2. The perimeter of the base is 5 + 12 + 13 = 30 cm. The three lateral rectangular faces sum to Perimeter * Length = 30 * 20 = 600 cm^2. Total Surface Area = 60 + 600 = 660 square centimeters.

Solution to Exercise 3. The cereal box is a rectangular prism with l = 8 in., w = 2.5 in., and h = 12 in. SA = 2(lw + lh + wh) = 2(8 * 2.5 + 8 * 12 + 2.5 * 12) = 2(20 + 96 + 30) = 2(146) = 292 square inches of cardboard.

Solution to Exercise 4. Because the chest has an open top, we calculate 1 bottom face, 2 front/back faces, and 2 side faces: Bottom = l * w = 4 * 2.5 = 10 ft^2. Front and Back = 2 * (l * h) = 2 * (4 * 2) = 16 ft^2. Left and Right sides = 2 * (w * h) = 2 * (2.5 * 2) = 10 ft^2. Total painted surface area = 10 + 16 + 10 = 36 square feet.