The lateral surface area is sum of the areas of the rectangular shaped sides of the prism or the areas of the triangles of the pyramid. It does NOT include the area of the bases.

Answer the following questions. Click on the blank space to check your answers.

• Describe the difference between total surface area and lateral surface area. Interactive popup. Assistance may be required.

  _______________

  The total surface area is the sum of the areas of all the faces. Lateral surface area is the sum of only the lateral faces, excluding the base(s).

• If the formula for the total area of a rectangular prism is S = Ph + 2B, what is the formula for the lateral area of a rectangular prism? Interactive popup. Assistance may be required.

  _______________

  S = Ph

The formula for lateral surface area of a rectangular prism will be the total surface area formula without the areas of the bases.

Example: Given the rectangular prism below, find the lateral surface area.

• The perimeter (P) of the base of the rectangular prism is Interactive popup. Assistance may be required.

  _______________

  10 + 3 + 10 + 3 = 26
  .
• The height (h) of the rectangular prism is Interactive popup. Assistance may be required.

  _______________

  5

• The lateral area of the above prism is Interactive popup. Assistance may be required.

  _______________

  S = 26(5) = 130 units²

The formula for lateral surface area of a pyramid will be the total surface area formula without the area of the base or

• Pyramids have one Interactive popup. Assistance may be required. _______________ polygonal base.
• The lateral faces of a pyramid are in the shape of Interactive popup. Assistance may be required.

  _______________

  triangles

• The Interactive popup. Assistance may be required. _______________ lateral surface area is the sum of the areas of the triangular lateral faces, without the area of the base.
• The formula for the lateral surface area of a pyramid is Interactive popup. Assistance may be required.

  _______________

  S = 1 2 Pl

Example: Given the pyramid below find the lateral area.

• The perimeter (P) of the pyramid is _______________ 7 + 7 + 7 + 7 = 28
  .
• The slant height (l) of the pyramid is _______________ 10
  .
• The lateral area of the above pyramid is _______________ S = 1 2 (28)10 = 140 units²

You may also encounter problems that ask you to find the surface area of only certain faces of a prism or pyramid-like object. Due to practical limitations, such as hidden faces or holes in the object, you may have to use formulas, then get a little creative.

For example: An obelisk is a 3-dimensional object with a square base, lateral faces that are trapezoids, and a crown that is a square pyramid. A famous obelisk is the Washington Monument, in Washington, DC.

Given these dimensions, what is the total surface area of all 8 exposed faces of the Washington Monument?

Answer the following questions. Click on the blank space to check your answers.

Step 1: Determine the total area of the lateral faces of the trapezoidal portion of the bottom of the obelisk.

• What is the formula for the area of a trapezoid? Interactive popup. Assistance may be required.

  _______________

  1 2 (b₁ + b₂)h

• Substitute the values you know into the area formula.

  _______________ 1 2 (55 + 34.5)(500) = 1 2 (89.5)(500) = 22,375 square feet

• How many trapezoids are there for the bottom portion of the obelisk? Interactive popup. Assistance may be required.

  _______________

  4

• What is the total area of the lateral faces of the bottom portion of the obelisk? Interactive popup. Assistance may be required.

  _______________

  4(22,375) = 89,500 square feet

Step 2: Determine the lateral surface area of the pyramid on top of the obelisk.

• What is the formula for the lateral surface area of a pyramid? Interactive popup. Assistance may be required.

  _______________

  A = 1 2 Pl

• What is the side length of the base of the pyramid at the top of the obelisk? Interactive popup. Assistance may be required.

  _______________

  34.5 feet

• What is the perimeter of the base of the pyramid at the top of the obelisk? Interactive popup. Assistance may be required.

  _______________

  138 feet

Use the Pythagorean Theorem to determine the slant height of the pyramid.

What is the height of the pyramid (length of the vertical leg of the right triangle?) Interactive popup. Assistance may be required. _______________ 50 feet
What is the length of the horizontal leg of the right triangle? Interactive popup. Assistance may be required. _______________ 1 2 (34.5 feet) = 17.25 feet
Using the Pythagorean Theorem, what is the slant height? Interactive popup. Assistance may be required. _______________ a2 + b2 = c2 (17.25)2 + (50)2 = l2 2798 ≈ l2 53 ≈ l

• Determine the lateral surface area of the pyramid at the top of the obelisk using the formula. Interactive popup. Assistance may be required.

  _______________

  A = 1 2 Pl ≈ 1 2 (138)(53)≈3,657 square feet

Step 3: Combine the areas of the two portions of the obelisk.

• Add the surface area of the bottom portion to the surface area of the top portion of the obelisk. Interactive popup. Assistance may be required.

  _______________

  89,500 square feet + 3,657 square feet = 93,157 square feet