Surface Area of Rectangular and Triangular Prisms - Seventh Grade Mathematics

Surface area measures the total two-dimensional boundary enclosing a three-dimensional prism, calculated by unfolding solids into planar geometric nets and summing face areas.

Deconstructing 3D Solids into 2D Nets

The surface area of a three-dimensional solid is the total sum of the areas of all its outer faces, measured in square units. A geometric net is a two-dimensional unfolding of a solid that can be folded along its edges to recreate the three-dimensional figure.

               NET OF A RECTANGULAR PRISM
                         +-------+
                         |  Top  |
                 +-------+-------+-------+-------+
                 | Left  | Front | Right | Back  |
                 +-------+-------+-------+-------+
                         |Bottom |
                         +-------+

Unfolding a solid into a flat net ensures that every face is accounted for, eliminating the risk of omitting faces or double-counting areas.

Surface Area of Rectangular Prisms

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

               RECTANGULAR PRISM SURFACE AREA
   SA = 2 * (l * w) + 2 * (l * h) + 2 * (w * h)
   where l = length, w = width, h = height

Alternatively, surface area can be expressed in terms of the base area B and the lateral area LA: The perimeter of the base is P equals two l plus two w. The lateral area is the perimeter of the base times the height: LA equals P times h. The total surface area is twice the base area plus lateral area: SA equals two B plus P times h.

Example: Find surface area of a box with l = 8 cm, w = 5 cm, h = 3 cm.
Top & Bottom: 2 * (8 * 5) = 2 * 40 = 80 sq cm
Front & Back: 2 * (8 * 3) = 2 * 24 = 48 sq cm
Left & Right: 2 * (5 * 3) = 2 * 15 = 30 sq cm
Total Surface Area = 80 + 48 + 30 = 158 square centimeters

Surface Area of Triangular Prisms

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

               NET OF A TRIANGULAR PRISM
                         /\  (Triangular Base 1)
                        /                   +-----+----+-----+
                 | Rec |Rec | Rec |  <-- 3 Rectangular Lateral Faces
                 |  1  | 2  |  3  |
                 +-----+----+-----+
                        \  /
                         \/  (Triangular Base 2)

To calculate the surface area of a triangular prism: Step 1: Calculate the area of the two triangular bases: two times the quantity one-half base times height of the triangle. The factor of two and one-half cancel, leaving base times height (b * h_tri). Step 2: Calculate the area of the three rectangular lateral faces: each rectangle has a length equal to one side of the triangle (s1, s2, s3) and a height equal to the prism length H: lateral area equals the quantity s1 plus s2 plus s3 times H, which is base perimeter P times H. Step 3: Sum the bases and lateral area: SA equals b times h_tri plus P times H.

Crucial Distinction:
  h_tri = Perpendicular height of the TRIANGULAR base
  H     = Height (length) of the PRISM separating the two bases

Conflating the height of the triangular face with the overall length of the prism is the primary source of computational error in triangular prism problems.

Exercises

Work through each problem carefully in your notebook before reading the provided solutions.

Exercise 1. Calculate the total surface area of a rectangular cardboard shipping box measuring ten inches long, six inches wide, and four inches high.

Exercise 2. A cube has edge lengths measuring seven centimeters each. Determine the total surface area of the cube.

Exercise 3. A triangular prism has right-triangular bases with legs measuring three meters and four meters, and a hypotenuse measuring five meters. The length of the prism between the two bases is ten meters. Calculate the total surface area.

Exercise 4. An open-top wooden planter box is constructed in the shape of a rectangular prism measuring twelve inches long, eight inches wide, and six inches deep. Because the box has no top lid, calculate the total exterior surface area of the wood.

Solutions

Solution to Exercise 1. Use the rectangular prism formula SA equals two lw plus two lh plus two wh with length ten, width six, and height four. Calculate top and bottom: two times ten times six equals two times sixty, which is one hundred twenty. Calculate front and back: two times ten times four equals two times forty, which is eighty. Calculate two ends: two times six times four equals two times twenty-four, which is forty-eight. Sum the areas: one hundred twenty plus eighty plus forty-eight equals two hundred forty-eight square inches.

Solution to Exercise 2. A cube has six identical square faces. The area of one square face is side length squared: seven squared equals forty-nine square centimeters. Multiply the area of one face by six: six times forty-nine equals two hundred ninety-four square centimeters. The total surface area of the cube is two hundred ninety-four square centimeters.

Solution to Exercise 3. First calculate the area of the two right-triangular bases. The base and height of the right triangle are the two legs, three meters and four meters. Area of one triangle equals one-half times three times four, which is six square meters. Two bases have a combined area of two times six equals twelve square meters. Next calculate the perimeter of the triangular base: three plus four plus five equals twelve meters. Multiply the perimeter by the prism length H of ten meters to find the lateral area: twelve times ten equals one hundred twenty square meters. Finally, sum base area and lateral area: twelve plus one hundred twenty equals one hundred thirty-two square meters.

Solution to Exercise 4. An open-top box has five faces: one bottom base and four lateral sides (front, back, left, and right). The top face is excluded. Area of bottom base: twelve times eight equals ninety-six square inches. Area of front and back faces: two times twelve times six equals two times seventy-two, which is one hundred forty-four square inches. Area of left and right faces: two times eight times six equals two times forty-eight, which is ninety-six square inches. Sum the five faces: ninety-six plus one hundred forty-four plus ninety-six equals three hundred thirty-six square inches.