Lesson 22: Surface Area

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 7•3

Student Outcomes . Students find the surface area of three-dimensional objects whose surface area is composed of triangles and quadrilaterals, specifically focusing on pyramids. 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 Notes Before class, teachers need to make copies of the composite figure for the Opening Exercise on cardstock. To save class time, they could also cut these nets out for students.

Classwork Opening Exercise (5 minutes) Make copies of the composite figure on cardstock and have students cut and fold the net to form the three-dimensional object.

Opening Exercise

What is the area of the composite figure in the diagram? Is the diagram a net for a three-dimensional image? If so, sketch the image. If not, explain why.

There are four unit squares in each square of the figure. There are ퟏퟖ total squares that make up the figure, so the total area of the composite figure is

푨 = ퟏퟖ ∙ ퟒ 퐮퐧퐢퐭퐬ퟐ = ퟕퟐ 퐮퐧퐢퐭퐬ퟐ.

The composite figure does represent the net of a three-dimensional figure. The figure is shown below.

Lesson 22: Surface Area 308

This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org This work is licensed under a This file derived from G7-M3-TE-1.3.0-08.2015 Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 7•3

Example 1 (5 minutes)

Pyramids are formally defined and explored in more depth in Module 6. Here, we simply introduce finding the surface area of a pyramid and ask questions designed to elicit the formulas from students. For example, ask how many lateral faces there are on the pyramid; then, ask for the triangle area formula. Continue leading students toward stating the formula for total surface area on their own.

Example 1

The pyramid in the picture has a square base, and its lateral faces are triangles that are exact copies of one another. Find the surface area of the pyramid.

The surface area of the pyramid consists of one square base and four lateral triangular faces. ퟏ 푳푨 = ퟒ ( 풃풉) 푩 = 풔ퟐ ퟐ ퟏ 푳푨 = ퟒ ∙ (ퟔ 퐜퐦 ∙ ퟕ 퐜퐦) 푩 = (ퟔ 퐜퐦)ퟐ ퟐ 푳푨 = ퟐ(ퟔ 퐜퐦 ∙ ퟕ 퐜퐦) 푩 = ퟑퟔ 퐜퐦ퟐ

푳푨 = ퟐ(ퟒퟐ 퐜퐦ퟐ) The pyramid’s base area is ퟑퟔ 퐜퐦ퟐ.

푳푨 = ퟖퟒ 퐜퐦ퟐ

The pyramid’s lateral area is ퟖퟒ 퐜퐦ퟐ. 푺푨 = 푳푨 + 푩

푺푨 = ퟖퟒ 퐜퐦ퟐ + ퟑퟔ 퐜퐦ퟐ 푺푨 = ퟏퟐퟎ 퐜퐦ퟐ

The surface area of the pyramid is ퟏퟐퟎ 퐜퐦ퟐ.

Example 2 (4 minutes): Using Cubes

Consider providing 13 interlocking cubes to small groups of students so they may construct a model of the diagram shown. Remind students to count faces systematically. For example, first consider only the bottom 9 cubes. This structure has a surface area of 30 (9 at the top, 9 at the bottom, and 3 on each of the four sides). Now consider the four cubes added at the top. Since we have already counted the tops of these cubes, we just need to add the four sides of 1 each. 30 + 16 = 46 total square faces, each with side length inch. 4

Example 2: Using Cubes ퟏ ퟏퟑ There are cubes glued together forming the solid in the diagram. The edges of each cube are ퟒ inch in length. Find the surface area of the solid. ퟏ ퟒퟔ The surface area of the solid consists of square faces, all having side lengths of ퟒ inch. The area of a square having ퟏ ퟏ 퐢퐧ퟐ sides of length ퟒ inch is ퟏퟔ .

푺푨 = ퟒퟔ ∙ 푨퐬퐪퐮퐚퐫퐞 ퟏ 푺푨 = ퟒퟔ ∙ 퐢퐧ퟐ ퟏퟔ ퟒퟔ 푺푨 = 퐢퐧ퟐ ퟏퟔ ퟏퟒ 푺푨 = ퟐ 퐢퐧ퟐ ퟏퟔ ퟕ ퟕ 푺푨 = ퟐ 퐢퐧ퟐ ퟐ 퐢퐧ퟐ. ퟖ The surface are of the solid is ퟖ

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 7•3

. Compare and contrast the methods for finding surface area for pyramids and prisms. How are the methods similar?  When finding the surface area of a pyramid and a prism, we find the area of each face and then add these areas together. . How are they different?  Calculating the surface area of pyramids and prisms is different because of the shape of the faces. A prism has two bases and lateral faces that are rectangles. A pyramid has only one base and lateral faces that are triangles.

Example 3 (15 minutes)

Example 3 ퟏ

Find the total surface area of the wooden jewelry box. The sides and bottom of the box are all ퟒ inch thick. What are the faces that make up this box?

The box has a rectangular bottom, rectangular lateral faces, and a rectangular top that has a smaller rectangle removed from it. There are also rectangular faces that make up the inner lateral faces and the inner bottom of the box.

How does this box compare to other objects that you have found the surface area of?

The box is a rectangular prism with a smaller rectangular prism removed from its inside. The total surface area is equal to the surface area of the larger right rectangular prism plus the lateral area of the smaller right rectangular prism.

ퟏ 퐢퐧 Large Prism: The surface area of the large right rectangular Small Prism: The smaller prism is ퟐ . smaller in prism makes up the outside faces of the box, the rim of the box, ퟏ length and width and 퐢퐧. smaller in height due to and the inside bottom face of the box. ퟒ the thickness of the sides of the box.

푺푨 = 푳푨 + ퟐ푩 푺푨 = 푳푨 + ퟏ푩 푳푨 = 푷 ∙ 풉 푩 = 풍풘 푳푨 = 푷 ∙ 풉 푳푨 = ퟑퟐ 퐢퐧.∙ ퟒ 퐢퐧. 푩 = ퟏퟎ 퐢퐧.∙ ퟔ 퐢퐧. ퟏ ퟏ ퟑ ퟐ ퟐ 푳푨 = ퟐ (ퟗ 퐢퐧. +ퟓ 퐢퐧. ) ∙ ퟑ 퐢퐧. 푳푨 = ퟏퟐퟖ 퐢퐧 푩 = ퟔퟎ 퐢퐧 ퟐ ퟐ ퟒ The lateral area is ퟏퟐퟖ 퐢퐧ퟐ. ퟑ ퟐ 푳푨 = ퟐ(ퟏퟒ 퐢퐧. +ퟏ 퐢퐧. ) ∙ ퟑ 퐢퐧. The base area is ퟔퟎ 퐢퐧 . ퟒ ퟑ 푳푨 = ퟐ(ퟏퟓ 퐢퐧. ) ∙ ퟑ 퐢퐧. 푺푨 = 푳푨 + ퟐ푩 ퟒ 푺푨 = ퟏퟐퟖ 퐢퐧ퟐ + ퟐ(ퟔퟎ 퐢퐧ퟐ) ퟑ 푳푨 = ퟑퟎ 퐢퐧.∙ ퟑ 퐢퐧. 푺푨 = ퟏퟐퟖ 퐢퐧ퟐ + ퟏퟐퟎ 퐢퐧ퟐ ퟒ 푺푨 = ퟐퟒퟖ 퐢퐧ퟐ ퟗퟎ 푳푨 = ퟗퟎ 퐢퐧ퟐ + 퐢퐧ퟐ The surface area of the larger prism is ퟐퟒퟖ 퐢퐧ퟐ. ퟒ ퟏ 푳푨 = ퟗퟎ 퐢퐧ퟐ + ퟐퟐ 퐢퐧ퟐ ퟐ Surface Area of the Box ퟏ 푳푨 = ퟏퟏퟐ 퐢퐧ퟐ 푺푨 = 푺푨 + 푳푨 ퟐ 퐛퐨퐱 ퟏ ퟏ ퟏퟏퟐ 퐢퐧ퟐ 푺푨 = ퟐퟒퟖ 퐢퐧ퟐ + ퟏퟏퟐ 풊풏ퟐ The lateral area is ퟐ . 풃풐풙 ퟐ ퟏ 푺푨 = ퟑퟔퟎ 퐢퐧ퟐ 풃풐풙 ퟐ ퟏ ퟑퟔퟎ 퐢퐧ퟐ The total surface area of the box is ퟐ .

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 7•3

Discussion (5 minutes): Strategies and Observations from Example 3

Call on students to provide their answers to each of the following questions. To Scaffolding: encourage a discussion about strategy, patterns, arguments, or observations, call on more To help students visualize the than one student for each of these questions. various faces involved on this . What ideas did you have to solve this problem? Explain. object, consider constructing a  Answers will vary. similar object by placing a smaller shoe box inside a . Did you make any mistakes in your solution? Explain. slightly larger shoe box. This  Answers will vary; examples include the following: will also help students visualize 1 1 the inner surfaces of the box as . I subtracted inch from the depth of the box instead of inch; 2 4 the lateral faces of the smaller 1 . I subtracted only inch from the length and width because I prism that is removed from the 4 larger prism. didn’t account for both sides. . Describe how you found the surface area of the box and what that surface area is.  Answers will vary.

Closing (2 minutes) . What are some strategies for finding the surface area of solids?  Answers will vary but might include creating nets, adding the areas of polygonal faces, and counting square faces and adding their areas.

Exit Ticket (9 minutes)

Lesson 22: Surface Area 311

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 7•3

Name Date

Lesson 22: Surface Area

Exit Ticket

1. The right hexagonal pyramid has a hexagon base with equal-length sides. The lateral faces of the pyramid are all triangles (that are exact copies of one another) with heights of 15 ft. Find the surface area of the pyramid.

2. Six cubes are glued together to form the solid shown in the 1 diagram. If the edges of each cube measure 1 inches in length, 2 what is the surface area of the solid?

Lesson 22: Surface Area 312

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 7•3

Exit Ticket Sample Solutions

푺푨 = 푳푨 + ퟏ푩 ퟏ 푳푨 = ퟔ ∙ (풃풉) 푩 = 푨 + ퟐ푨 ퟐ rectangle triangle ퟏ ퟏ 푳푨 = ퟔ ∙ (ퟓ 퐟퐭.∙ ퟏퟓ 퐟퐭. ) 푩 = (ퟖ 풇풕.∙ ퟓ 풇풕. ) + ퟐ ∙ (ퟖ 풇풕.∙ ퟑ 풇풕. ) ퟐ ퟐ 푳푨 = ퟑ ∙ ퟕퟓ 퐟퐭ퟐ 푩 = ퟒퟎ 퐟퐭ퟐ + (ퟖ 퐟퐭.∙ ퟑ 퐟퐭. )

푳푨 = ퟐퟐퟓ 퐟퐭ퟐ 푩 = ퟒퟎ 퐟퐭ퟐ + ퟐퟒ 퐟퐭ퟐ

푩 = ퟔퟒ 퐟퐭ퟐ 푺푨 = 푳푨 + ퟏ푩 푺푨 = ퟐퟐퟓ 퐟퐭ퟐ + ퟔퟒ 퐟퐭ퟐ

푺푨 = ퟐퟖퟗ 퐟퐭ퟐ

The surface area of the pyramid is ퟐퟖퟗ 퐟퐭ퟐ.

ퟏ ퟏ 2. Six cubes are glued together to form the solid shown in the diagram. If the edges of each cube measure ퟐ inches in length, what is the surface area of the solid? There are ퟐퟔ square cube faces showing on the surface area of the solid (ퟓ each from the top and bottom view, ퟒ each from the front and back view, ퟑ each from the left and right side views, and ퟐ from the “inside” of the front). ퟏ 푨 = 풔ퟐ 푺푨 = ퟐퟔ ∙ (ퟐ 퐢퐧ퟐ) ퟒ ퟏ ퟐ ퟐퟔ 푨 = (ퟏ 퐢퐧. ) 푺푨 = ퟓퟐ 퐢퐧ퟐ + 퐢퐧ퟐ ퟐ ퟒ ퟏ ퟏ ퟏ 푨 = (ퟏ 퐢퐧. ) (ퟏ 퐢퐧. ) 푺푨 = ퟓퟐ 퐢퐧ퟐ + ퟔ 퐢퐧ퟐ + 퐢퐧ퟐ ퟐ ퟐ ퟐ ퟏ ퟏ ퟏ 푨 = ퟏ 퐢퐧. (ퟏ 퐢퐧. + 퐢퐧. ) 푺푨 = ퟓퟖ 퐢퐧ퟐ ퟐ ퟐ ퟐ ퟏ ퟏ ퟏ 푨 = (ퟏ 퐢퐧.∙ ퟏ 퐢퐧. ) + (ퟏ 퐢퐧.∙ 퐢퐧. ) ퟐ ퟐ ퟐ ퟏ ퟑ 푨 = ퟏ 퐢퐧ퟐ + 퐢퐧ퟐ ퟐ ퟒ ퟐ ퟑ 푨 = ퟏ 퐢퐧ퟐ + 퐢퐧ퟐ ퟒ ퟒ ퟓ ퟏ ퟏ 푨 = ퟏ 퐢퐧ퟐ = ퟐ 퐢퐧ퟐ ퟓퟖ 퐢퐧ퟐ ퟒ ퟒ The surface area of the solid is ퟐ .

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 7•3

Problem Set Sample Solutions

1. For each of the following nets, draw (or describe) the solid represented by the net and find its surface area. a. The equilateral triangles are exact copies. The net represents a triangular pyramid where the three lateral faces are identical to each other and the triangular base. 푺푨 = ퟒ푩 since the faces are all the same size and shape. ퟏ 푩 = 풃풉 푺푨 = ퟒ푩 ퟐ ퟏ ퟒ ퟏ 푩 = ∙ ퟗ 퐦퐦 ∙ ퟕ 퐦퐦 푺푨 = ퟒ (ퟑퟓ 퐦퐦ퟐ) ퟐ ퟓ ퟏퟎ ퟗ ퟒ ퟒ 푩 = 퐦퐦 ∙ ퟕ 퐦퐦 푺푨 = ퟏퟒퟎ 퐦퐦ퟐ + 퐦퐦ퟐ ퟐ ퟓ ퟏퟎ ퟔퟑ ퟑퟔ ퟐ 푩 = 퐦퐦ퟐ + 퐦퐦ퟐ 푺푨 = ퟏퟒퟎ 퐦퐦ퟐ ퟐ ퟏퟎ ퟓ ퟑퟏퟓ ퟑퟔ 푩 = 퐦퐦ퟐ + 퐦퐦ퟐ ퟏퟎ ퟏퟎ ퟑퟓퟏ ퟐ 푩 = 퐦퐦ퟐ ퟏퟒퟎ 퐦퐦ퟐ ퟏퟎ The surface area of the triangular pyramid is ퟓ . ퟏ 푩 = ퟑퟓ 퐦퐦ퟐ ퟏퟎ

b. The net represents a square pyramid that has four identical lateral faces that are triangles. The base is a square.

푺푨 = 푳푨 + 푩 ퟏ 푳푨 = ퟒ ∙ (풃풉) 푩 = 풔ퟐ ퟐ ퟏ ퟑ 푳푨 = ퟒ ∙ (ퟏퟐ 퐢퐧.∙ ퟏퟒ 퐢퐧. ) 푩 = (ퟏퟐ 퐢퐧 )ퟐ ퟐ ퟒ . ퟑ 푳푨 = ퟐ (ퟏퟐ 퐢퐧.∙ ퟏퟒ 퐢퐧 ) 푩 = ퟏퟒퟒ 퐢퐧ퟐ ퟒ . 푳푨 = ퟐ(ퟏퟔퟖ 퐢퐧ퟐ + ퟗ 퐢퐧ퟐ)

푳푨 = ퟑퟑퟔ 퐢퐧ퟐ + ퟏퟖ 퐢퐧ퟐ

푳푨 = ퟑퟓퟒ 퐢퐧ퟐ 푺푨 = 푳푨 + 푩 푺푨 = ퟑퟓퟒ 퐢퐧ퟐ + ퟏퟒퟒ 퐢퐧ퟐ

푺푨 = ퟒퟗퟖ 퐢퐧ퟐ

The surface area of the square pyramid is ퟒퟗퟖ 퐢퐧ퟐ.

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 7•3

2. Find the surface area of the following prism.

푺푨 = 푳푨 + ퟐ푩 푳푨 = 푷 ∙ 풉 ퟏ ퟏ ퟏ 푳푨 = (ퟒ 퐜퐦 + ퟔ 퐜퐦 + ퟒ 퐜퐦 + ퟓ 퐜퐦) ∙ ퟗ 퐜퐦 ퟐ ퟓ ퟒ ퟏ ퟏ ퟏ 푳푨 = (ퟏퟗ 퐜퐦 + 퐜퐦 + 퐜퐦 + 퐜퐦) ∙ ퟗ 퐜퐦 ퟐ ퟓ ퟒ ퟏퟎ ퟒ ퟓ 푳푨 = (ퟏퟗ 퐜퐦 + 퐜퐦 + 퐜퐦 + 퐜퐦) ∙ ퟗ 퐜퐦 ퟐퟎ ퟐퟎ ퟐퟎ ퟏퟗ 푳푨 = (ퟏퟗ 퐜퐦 + 퐜퐦) ∙ ퟗ 퐜퐦 ퟐퟎ ퟏퟕퟏ 푳푨 = ퟏퟕퟏ 퐜퐦ퟐ + 퐜퐦ퟐ ퟐퟎ ퟏퟏ 푳푨 = ퟏퟕퟏ 퐜퐦ퟐ + ퟖ 퐜퐦ퟐ ퟐퟎ ퟏퟏ 푳푨 = ퟏퟕퟗ 퐜퐦ퟐ ퟐퟎ

푩 = 푨퐫퐞퐜퐭퐚퐧퐠퐥퐞 + 푨퐭퐫퐢퐚퐧퐠퐥퐞 푺푨 = 푳푨 + ퟐ푩 ퟏ ퟏ ퟏ ퟏퟏ ퟏ 푩 = (ퟓ 퐜퐦 ∙ ퟒ 퐜퐦) + (ퟒ 퐜퐦 ∙ ퟏ 퐜퐦) 푺푨 = ퟏퟕퟗ 퐜퐦ퟐ + ퟐ (ퟐퟑ 퐜퐦ퟐ) ퟒ ퟐ ퟒ ퟐퟎ ퟐ ퟏ ퟏퟏ 푩 = (ퟐퟎ 퐜퐦ퟐ + ퟏ 퐜퐦ퟐ) + (ퟐ 퐜퐦 ∙ ퟏ 퐜퐦) 푺푨 = ퟏퟕퟗ 퐜퐦ퟐ + ퟒퟕ 퐜퐦ퟐ ퟒ ퟐퟎ ퟏ ퟏퟏ 푩 = ퟐퟏ 퐜퐦ퟐ + ퟐ 퐜퐦ퟐ 퐒퐀 = ퟐퟐퟔ 퐜퐦ퟐ ퟐ ퟐퟎ ퟏ 푩 = ퟐퟑ 퐜퐦ퟐ ퟐ ퟏퟏ ퟐퟐퟔ 퐜퐦ퟐ The surface area of the prism is ퟐퟎ .

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 7•3

3. The net below is for a specific object. The measurements shown are in meters. Sketch (or describe) the object, and then find its surface area.

(3-Dimensional Form)

푺푨 = 푳푨 + ퟐ푩 ퟏ ퟏ ퟏ ퟏ ퟏ 푳푨 = 푷 ∙ 풉 푩 = ( 퐜퐦 ∙ 퐜퐦) + ( 퐜퐦 ∙ ퟏ 퐜퐦) + ( 퐜퐦 ∙ ퟏ 퐜퐦) 푺푨 = 푳푨 + ퟐ푩 ퟐ ퟐ ퟐ ퟐ ퟐ ퟏ ퟏ ퟏ ퟑ ퟏ 푳푨 = ퟔ 퐜퐦 ∙ 퐜퐦 푩 = ( 퐜퐦ퟐ) + ( 퐜퐦ퟐ) + ( 퐜퐦ퟐ) 푺푨 = ퟑ 퐜퐦ퟐ + ퟐ (ퟏ 퐜퐦ퟐ) ퟐ ퟒ ퟐ ퟒ ퟐ ퟏ ퟐ ퟑ 푳푨 = ퟑ 퐜퐦ퟐ 푩 = ( 퐜퐦ퟐ) + ( 퐜퐦ퟐ) + ( 퐜퐦ퟐ) 푺푨 = ퟑ 퐜퐦ퟐ + ퟑ 퐜퐦ퟐ ퟒ ퟒ ퟒ ퟔ 푩 = 퐜퐦ퟐ 푺푨 = ퟔ 퐜퐦ퟐ ퟒ ퟏ 푩 = ퟏ 퐜퐦ퟐ ퟐ The surface area of the object is ퟔ 퐜퐦ퟐ.

ퟏ ퟏퟒ 퐢퐧ퟑ 4. In the diagram, there are cubes glued together to form a solid. Each cube has a volume of ퟖ . Find the surface area of the solid.

ퟑ ퟏ ퟏ 풔ퟑ 퐢퐧ퟑ ( 퐢퐧. ) The volume of a cube is , and ퟖ is the same as ퟐ , so the ퟏ cubes have edges that are 퐢퐧. long. The cube faces have area 풔ퟐ, or ퟐ ퟏ ퟐ ퟏ ( 퐢퐧. ) , or 퐢퐧ퟐ. There are ퟒퟐ cube faces that make up the surface ퟐ ퟒ of the solid. ퟏ 푺푨 = 퐢퐧ퟐ ∙ ퟒퟐ ퟒ ퟏ 푺푨 = ퟏퟎ 퐢퐧ퟐ ퟐ ퟏ ퟏퟎ 퐢퐧ퟐ The surface area of the solid is ퟐ .

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NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 7•3

5. The nets below represent three solids. Sketch (or describe) each solid, and find its surface area. a. b. c.

푺푨 = 푳푨 + ퟐ푩 푺푨 = ퟑ푨퐬퐪퐮퐚퐫퐞 + ퟑ푨퐫퐭 퐭퐫퐢퐚퐧퐠퐥퐞 + 푨퐞퐪퐮 퐭퐫퐢퐚퐧퐠퐥퐞 푺푨 = ퟑ푨풓풕 풕풓풊풂풏품풍풆 + 푨풆풒풖 풕풓풊풂풏품풍풆 ퟐ 푳푨 = 푷 ∙ 풉 푨퐬퐪퐮퐚퐫퐞 = 풔 ퟏ ퟐ ퟕퟕ ퟐ 푳푨 = ퟏퟐ ∙ ퟑ ퟐ 푺푨 = ퟑ (ퟒ ) 퐜퐦 + ퟕ 퐜퐦 푨퐬퐪퐮퐚퐫퐞 = (ퟑ 퐜퐦) ퟐ ퟏퟎퟎ ퟐ 푳푨 = ퟑퟔ 풄풎 푨 = ퟗ 퐜퐦ퟐ ퟑ ퟕퟕ 퐬퐪퐮퐚퐫퐞 푺푨 = ퟏퟐ 퐜퐦ퟐ + 퐜퐦ퟐ + ퟕ 퐜퐦ퟐ + 퐜퐦ퟐ ퟏ ퟐ ퟏퟎퟎ ퟐ 푨퐫퐭 퐭퐫퐢퐚퐧퐠퐥퐞 = 풃풉 ퟏ ퟕퟕ 푩 = 풔 ퟐ 푺푨 = ퟐퟎ 퐜퐦ퟐ + 퐜퐦ퟐ + 퐜퐦ퟐ ퟐ ퟏퟎퟎ 푩 = (ퟑ 퐜퐦)ퟐ ퟏ 푨 = ∙ ퟑ 퐜퐦 ∙ ퟑ 퐜퐦 퐫퐭 퐭퐫퐢퐚퐧퐠퐥퐞 ퟐ ퟐ ퟐퟕ ퟐ 푩 = ퟗ 퐜퐦ퟐ ퟐ 푺푨 = ퟐퟎ 퐜퐦 + ퟏ 퐜퐦 + 퐜퐦 ퟗ ퟏퟎퟎ 푨퐫퐭 퐭퐫퐢퐚퐧퐠퐥퐞 = ퟐퟕ ퟐ 푺푨 = ퟐퟏ 퐜퐦2 푺푨 = ퟏ ퟏퟎퟎ ퟐ ퟐ 푨 = ퟒ 퐜퐦2 ퟑퟔ 퐜퐦 + ퟐ(ퟗ 퐜퐦 ) 퐫퐭 퐭퐫퐢퐚퐧퐠퐞 ퟐ 푺푨 = ퟑퟔ 퐜퐦ퟐ + ퟏퟖ 퐜퐦ퟐ ퟏ Sketch A 푨퐞퐪퐮 퐭퐫퐢퐚퐧퐠퐥퐞 = 풃풉 푺푨 = ퟓퟒ 퐜퐦ퟐ ퟐ ퟏ ퟏ ퟕ 푨 = ∙ (ퟒ 퐜퐦) ∙ (ퟑ 퐜퐦) 퐞퐪퐮 퐭퐫퐢퐚퐧퐠퐥퐞 ퟐ ퟓ ퟏퟎ ퟏ ퟕ 푨 = ퟐ 퐜퐦 ∙ ퟑ 퐜퐦 퐞퐪퐮 퐭퐫퐢퐚퐧퐠퐥퐞 ퟏퟎ ퟏퟎ ퟐퟏ ퟑퟕ 푨 = 퐜퐦 ∙ 퐜퐦 퐞퐪퐮 퐭퐫퐢퐚퐧퐠퐥퐞 ퟏퟎ ퟏퟎ ퟕퟕퟕ 푨 = 퐜퐦ퟐ 퐞퐪퐮 퐭퐫퐢퐚퐧퐠퐥퐞 ퟏퟎퟎ Sketch B ퟕퟕ 푨 = ퟕ 퐜퐦ퟐ 퐞퐪퐮 퐭퐫퐢퐚퐧퐠퐥퐞 ퟏퟎퟎ ퟏ ퟕퟕ 푺푨 = ퟑ(ퟗ 퐜퐦ퟐ) + ퟑ (ퟒ 퐜퐦ퟐ) + ퟕ 퐜퐦ퟐ ퟐ ퟏퟎퟎ ퟑ ퟕퟕ 푺푨 = ퟐퟕ 퐜퐦ퟐ + (ퟏퟐ + ) 퐜퐦ퟐ + ퟕ 퐜퐦ퟐ ퟐ ퟏퟎퟎ ퟏ ퟕퟕ 푺푨 = ퟒퟕ 퐜퐦ퟐ + 퐜퐦ퟐ + 퐜퐦ퟐ ퟐ ퟏퟎퟎ Sketch C ퟓퟎ ퟕퟕ 푺푨 = ퟒퟕ 퐜퐦ퟐ + 퐜퐦ퟐ + 퐜퐦ퟐ ퟏퟎퟎ ퟏퟎퟎ ퟏퟐퟕ 푺푨 = ퟒퟕ 퐜퐦ퟐ + 퐜퐦ퟐ ퟏퟎퟎ ퟐퟕ 푺푨 = ퟒퟕ 퐜퐦ퟐ + ퟏ 퐜퐦ퟐ + 퐜퐦ퟐ ퟏퟎퟎ ퟐퟕ 푺푨 = ퟒퟖ 퐜퐦ퟐ ퟏퟎퟎ

Lesson 22: Surface Area 317

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 7•3

d. How are figures (b) and (c) related to figure (a)?

If the equilateral triangular faces of figures (b) and (c) were matched together, they would form the cube in part (a).

6. Find the surface area of the solid shown in the diagram. The solid is a right triangular prism (with right triangular bases) with a smaller right triangular prism removed from it.

푺푨 = 푳푨 + ퟐ푩 푳푨 = 푷 ∙ 풉 ퟏퟑ 푳푨 = (ퟒ 퐢퐧. +ퟒ 퐢퐧. +ퟓ 퐢퐧. ) ∙ ퟐ 퐢퐧. ퟐퟎ ퟏퟑ 푳푨 = (ퟏퟑ 퐢퐧. ) ∙ ퟐ 퐢퐧. ퟐퟎ ퟏퟑ 푳푨 = ퟐퟔ 퐢퐧ퟐ + 퐢퐧ퟐ ퟏퟎ ퟑ 푳푨 = ퟐퟔ 퐢퐧ퟐ + ퟏ 퐢퐧ퟐ + 퐢퐧ퟐ ퟏퟎ ퟑ 푳푨 = ퟐퟕ 퐢퐧ퟐ ퟏퟎ ퟏ ퟏퟗ The 퐢퐧. by ퟒ 퐢퐧. rectangle has to be taken away from the lateral area: ퟒ ퟐퟎ ퟑ ퟏퟗ 푨 = 풍풘 퐋퐀 = ퟐퟕ 퐢퐧ퟐ − ퟏ 퐢퐧ퟐ ퟏퟎ ퟖퟎ ퟏퟗ ퟏ ퟐퟒ ퟏퟗ 푨 = ퟒ 퐢퐧 ∙ 퐢퐧 퐋퐀 = ퟐퟕ 퐢퐧ퟐ − ퟏ 퐢퐧ퟐ ퟐퟎ ퟒ ퟖퟎ ퟖퟎ ퟏퟗ ퟓ 푨 = ퟏ 퐢퐧ퟐ + 퐢퐧ퟐ 푳푨 = ퟐퟔ 퐢퐧ퟐ ퟖퟎ ퟖퟎ ퟏퟗ ퟏ 푨 = ퟏ 퐢퐧ퟐ 푳푨 = ퟐퟔ 퐢퐧ퟐ ퟖퟎ ퟏퟔ

Two bases of the larger and smaller triangular prisms must be added: ퟏ ퟏ ퟏ ퟏ 푺푨 = ퟐퟔ 퐢퐧ퟐ + ퟐ (ퟑ 퐢퐧 ∙ 퐢퐧) + ퟐ ( ∙ ퟒ 퐢퐧 ∙ ퟒ 퐢퐧) ퟏퟔ ퟐ ퟒ ퟐ ퟏ ퟏ ퟏ 푺푨 = ퟐퟔ 퐢퐧ퟐ + ퟐ ∙ 퐢퐧 ∙ ퟑ 퐢퐧 + ퟏퟔ 퐢퐧ퟐ ퟏퟔ ퟒ ퟐ ퟏ ퟏ ퟏ 푺푨 = ퟐퟔ 퐢퐧ퟐ + 퐢퐧 ∙ ퟑ 퐢퐧 + ퟏퟔ 퐢퐧ퟐ ퟏퟔ ퟐ ퟐ ퟏ ퟑ ퟏ 푺푨 = ퟐퟔ 퐢퐧ퟐ + ( 퐢퐧ퟐ + 퐢퐧ퟐ) + ퟏퟔ 퐢퐧ퟐ ퟏퟔ ퟐ ퟒ ퟏ ퟖ ퟒ 푺푨 = ퟐퟔ 퐢퐧ퟐ + ퟏ 퐢퐧ퟐ + 퐢퐧ퟐ + 퐢퐧ퟐ + ퟏퟔ 퐢퐧ퟐ ퟏퟔ ퟏퟔ ퟏퟔ ퟏퟑ 푺푨 = ퟒퟑ 퐢퐧ퟐ ퟏퟔ ퟏퟑ ퟒퟑ 퐢퐧ퟐ The surface area of the solid is ퟏퟔ .

Lesson 22: Surface Area 318

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 22 7•3

7. The diagram shows a cubic meter that has had three square holes punched completely through the cube on three perpendicular axes. Find the surface area of the remaining solid.

Exterior surfaces of the cube (푺푨ퟏ): ퟏ ퟐ 푺푨 = ퟔ(ퟏ 퐦)ퟐ − ퟔ ( 퐦) ퟏ ퟐ ퟏ 푺푨 = ퟔ(ퟏ 퐦ퟐ) − ퟔ ( 퐦ퟐ) ퟏ ퟒ ퟔ 푺푨 = ퟔ 퐦ퟐ − 퐦ퟐ ퟏ ퟒ ퟏ 푺푨 = ퟔ 퐦ퟐ − (ퟏ 퐦ퟐ) ퟏ ퟐ ퟏ 푺푨 = ퟒ 퐦ퟐ ퟏ ퟐ Just inside each square hole are four intermediate surfaces that can be treated as the lateral area of a rectangular ퟏ ퟏ ퟏ ퟏ ퟏ prism. Each has a height of 퐦 and perimeter of 퐦 + 퐦 + 퐦 + 퐦 or ퟐ 퐦. ퟒ ퟐ ퟐ ퟐ ퟐ

푺푨ퟐ = ퟔ(푳푨) ퟏ 푺푨 = ퟔ (ퟐ 퐦 ∙ 퐦) ퟐ ퟒ ퟏ 푺푨 = ퟔ ∙ 퐦2 ퟐ ퟐ ퟐ 푺푨ퟐ = ퟑ 퐦

The total surface area of the remaining solid is the sum of these two areas:

푺푨푻 = 푺푨ퟏ + 푺푨ퟐ. ퟏ 푺푨 = ퟒ 퐦2 + ퟑ 퐦ퟐ 푻 ퟐ ퟏ 푺푨 = ퟕ 퐦ퟐ 푻 ퟐ ퟏ ퟕ 퐦ퟐ The surface area of the remaining solid is ퟐ .

Lesson 22: Surface Area 319