# Surface Area

Video solutions to help Grade 7 students learn how to find the surface area of three-dimensional objects whose surface area is composed of triangles and quadrilaterals.

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Lesson Plans and Worksheets for Grade 7

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## New York State Common Core Math Grade 7, Module 3, Lesson 21

### Lesson 21 Student Outcomes

• Students find the surface area of three-dimensional objects whose surface area is composed of triangles and quadrilaterals. They use polyhedron nets to understand that surface area is simply the sum of the area of the lateral faces and the area of the base(s).

Lesson 21 Classwork

Opening Exercise: Surface Area of a Right Rectangular Prism
On the provided grid, draw a net representing the surfaces of the right rectangular prism (assume each grid line represents inch). Then find the surface area of the prism by finding the area of the net.

Discussion: Terminology
A right prism can be described as a solid with two “end” faces (called its bases) that are exact copies of each other and rectangular faces that join corresponding edges of the bases (called lateral faces).

Exercise 1
Marcus thinks that the surface area of the right triangular prism will be half that of the right rectangular prism and wants to use the modified formula SA = 1/2(2lw + 2lh + 2wh). Do you agree or disagree with Marcus? Use nets of the prisms to support your argument.

Example 1
A right triangular prism, a right rectangular prism, and a right pentagonal prism are pictured below, and all have equal heights of h.
a. Write an expression that represents the lateral area of the right triangular prism as the sum of the areas of its lateral faces.
b. Write an expression that represents the lateral area of the right rectangular prism as the sum of the areas of its lateral faces.
c. Write an expression that represents the lateral area of the right pentagonal prism as the sum of the areas of its lateral faces.
d. What value appears often in each expression and why?
e. Rewrite each expression in factored form using the distributive property and the height of each lateral face.
f. What do the parentheses in each case represent with respect to the right prisms?
g. How can we generalize the lateral area of a right prism into a formula that applies to all right prisms?

Closing
Relevant Vocabulary
Right Prism: Let E and E' be two parallel planes. Let B be a triangular or rectangular region or a region that is the union of such regions in the plane E. At each point P of B, consider the segment perpendicular to E, joining P to a point P' of the plane E'. The union of all these segments is a solid called a right prism.

There is a region B' in E' that is an exact copy of the region B. The regions B and B' are called the base faces (or just bases) of the prism. The rectangular regions between two corresponding sides of the bases are called lateral faces of the prism. In all, the boundary of a right rectangular prism has faces: base faces and lateral faces. All adjacent faces intersect along segments called edges (base edges and lateral edges).

Cube: A cube is a right rectangular prism all of whose edges are of equal length.

Surface: The surface of a prism is the union of all of its faces (the base faces and lateral faces).

Net (description): A net is a two dimensional diagram of the surface of a prism.

1. Why are the lateral faces of right prisms always rectangular regions?
2. What is the name of the right prism whose bases are rectangles?
3. How does this definition of right prism include the interior of the prism?

Lesson 21 Summary The surface area of a right prism can be obtained by adding the areas of the lateral faces to the area of the bases. The formula for the surface area of a right prism is SA = LA + 2B, where SA represents surface area of the prism, LA represents the area of the lateral faces, and B represents the area of one base. The lateral area can be obtained by multiplying the perimeter of the base of the prism times the height of the prism.