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Illustrative Mathematics Unit 6.1, Lesson 14: Nets and Surface Area

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Learn about matching nets to their polyhedra and using nets to find the surface area of a polyhedron. After trying the questions, click on the buttons to view answers and explanations in text or video.

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Nets and Surface Area
Let’s use nets to find the surface area of polyhedra.

14.1 - Matching Nets

Each of the following nets can be assembled into a polyhedron. Match each net with its corresponding polyhedron, and name the polyhedron. Be prepared to explain how you know the net and polyhedron go together.

Five nets labeled A-E above five polyhedra labeled 1--5.

  • Answers

    A: 3 - Square pyramid, formed from a square base and 4 triangles.
    B: 2 - Square prism, formed from 2 square bases and 4 rectangles.
    C: 4 - Triangular pyramid (tetrahedron), formed from a triangular base and 3 triangles.
    D: 5 - Triangular prism, formed from 2 triangular bases and 3 rectangles.
    E: 1 - Cube, formed from 6 squares.

  • See Video 1 for Whole Lesson
  • See Video 2 for Whole Lesson




14.2 - Using Nets to Find Surface Area

Your teacher will give you the nets of three polyhedra to cut out and assemble. Alternatively, you may print out the following nets.

Three nets on a grid, labeled A, B, and C. Net A is composed of two rectangles that are 5 units tall by 6 units wide, two that are 5 units high and one unit wide, and two that are one unit high and six units wide. Net B is a square with a side length of 4 units and is surrounded by triangles that are four units wide at the base and four units high. Net C is a square with a side length of 3, a rectangle 3 units wide and 5 units high, another rectangle that is 3 units wide and 4 units tall, and two triangles, one on either side, that are three units tall by four units across.

1. Name the polyhedron that each net would form when assembled.

2. Cut out your nets and use them to create three-dimensional shapes.

3. Find the surface area of each polyhedron. Explain your reasoning clearly.

  • Hints

    1. Recall the definitions of a prism and a pyramid, and how these are named.

    3. A net allows us to see all the faces of a polygon at once. To find the surface area of a polyhedron from its net, find the total area of all the polygons in the net.

  • See Possible Answers

    1. A forms a rectangular prism. B forms a square pyramid. C forms a triangular prism (with right triangular bases).

    3. A net allows us to see all the faces of a polygon at once. To find the surface area of a polyhedron from its net, find the total area of all the polygons in the net. In each of these nets, polygons with the same area have been shaded the same color. To simplify calculations, polygons can also be combined to calculate area, as indicated by the dashed outlines in nets A and C.

    Three nets on a grid, labeled A, B, and C. Net A is composed of two rectangles that are 5 units tall by 6 units wide (shaded red), two that are 5 units high and one unit wide (shaded dark blue), and two that are one unit high and six units wide (blue). Net B is a square with a side length of 4 units (shaded red) and is surrounded by triangles that are four units wide at the base and four units high (blue). Net C is a square with a side length of 3 (blue), a rectangle 3 units wide and 5 units high (red), another rectangle that is 3 units wide and 4 units tall (dark blue), and two triangles (green), one on either side, that are three units tall by four units across.

    A: 2(5 × 6) + 2(6) + 2(5) = 82 square units
    OR (6 × 12) + 2(5) = 82 square units

    B: (4 × 4) + 4(½ × 4 × 4) = 48 square units

    C: (3 × 5) + (3 × 3) + (3 × 4) + 2(½ × 3 × 4) = 48 square units
    OR (3 × 12) + 2(½ × 3 × 4) = 48 square units

    Other strategies are also possible.


1. For each net, decide if it can be assembled into a rectangular prism.

Four possible nets labeled A--D.

2. For each net, decide if it can be assembled into a triangular prism.

Four possible nets labeled A--D.

  • Answers

    1. Only C can be folded into a rectangular prism.
    A has a square face which does not form a pair with any other face.
    B has a rectangle which contacts more than one other rectangle along one of its edges, and so cannot be folded into a polyhedron.
    D does not have enough faces to form a closed polyhedron.

    2. Only D can be folded into a triangular prism.
    A is impossible to fold such that the triangles are on opposite sides.
    B and C are impossible to fold such that all the sides of each face are in contact with another face.




Lesson 14 Summary

A net of a pyramid has one polygon that is the base. The rest of the polygons are triangles. A pentagonal pyramid and its net are shown here.

The net for this pentagonal pyramid is a pentagon surrounded by triangles on each side.

A net of a prism has two copies of the polygon that is the base. The rest of the polygons are rectangles. A pentagonal prism and its net are shown here.

The net for this pentagonal prism is a pentagon surrounded by rectangles on each side with an additional pentagon attached to the opposite side of one of the rectangles.

In a rectangular prism, there are three pairs of parallel and identical rectangles. Any pair of these identical rectangles can be the bases.

Three images of a rectangular prism. Each image has one set of opposing sides of the polyhedron shaded and labeled base.

Because a net shows all the faces of a polyhedron, we can use it to find its surface area.
For instance, the net of a rectangular prism shows three pairs of rectangles: 4 units by 2 units, 3 units by 2 units, and 4 units by 3 units.

A polyhedron made up of six rectangles. Two rectangles are 8 square units in area, 2 are 6 square units, and 2 are 12 square units.

The surface area of the rectangular prism is 52 square units because 8 + 8 + 6 + 6 + 12 + 12 = 52.



Practice Problems

1. Can the following net be assembled into a cube? Explain how you know. Label parts of the net with letters or numbers if it helps your explanation.

A net of 6 squares. 4 squares are lined up in a row, with 2 squares lined up in another row under the last square in the first row.

  • Answers

    A net of 6 squares. 4 squares are lined up in a row, with 2 squares lined up in another row under the last square in the first row. The squares have been labeled 1-6.

    This net cannot be assembled into a cube. Squares 1 and 6 will always overlap each other.


2. a. What polyhedron can be assembled from this net? Explain how you know.
b. Find the surface area of this polyhedron. Show your reasoning.

A net on a grid consisting of 2 triangles with base 4 and height 3, connected on either side to a rectangle 4 by 5. This rectangle is connected in a row with a second rectangle 3 by 5, and a square 5 by 5.

  • See Possible Answers

    a. This net can be assembled into a triangular prism. There are two congruent triangles which are the bases, and three rectangles for connecting the bases.

    b. Faces which have the same area have been shaded the same color. The contiguous rectangular faces can also be considered as one large rectangle to simplify calculations.

    A net on a grid consisting of 2 triangles with base 4 and height 3, connected on either side to a rectangle 4 by 5. This rectangle is connected in a row with a second rectangle 3 by 5, and a square 5 by 5.

    The area of this polyhedron is (5 × 4) + (5 × 3) + (5 × 5) + 2(½ × 3 × 4) = 72 square units
    OR (5 × 12) + 2(½ × 3 × 4) = 72 square units

    Other strategies are also possible.


3. Here are two nets. Mai said that both nets can be assembled into the same triangular prism. Do you agree? Explain or show your reasoning.

2 nets labeled A and B. A is composed of 2 triangular and 3 rectangular faces. B is composed of the same faces, but they are arranged differently.

  • Answers

    Mai is correct. In both nets, the rectangles can be folded to touch every edge of both triangular bases.


4. Here are two three-dimensional figures.

A triangular prism labeled A and a tetrahedron labeled B.

Tell whether each of the following statements describes Figure A, Figure B, both, or neither.

a. This figure is a polyhedron.
b. This figure has triangular faces.
c. There are more vertices than edges in this figure.
d. This figure has rectangular faces.
e. This figure is a pyramid.
f. There is exactly one face that can be the base for this figure.
g. The base of this figure is a triangle.
h. This figure has two identical and parallel faces that can be the base.

  • Answers

    a. Both - both figures are polyhedra.
    b. Both - Figure A is a triangular prism, and Figure B is a triangular pyramid.
    c. Neither - both figures have more edges than vertices.
    d. Figure A
    e. Figure B
    f. Neither - Figure A is a prism and has two bases. Any face of a triangular pyramid like Figure B can be its base.
    g. Both - Figure A is a triangular prism with two triangular bases, and Figure B is a triangular pyramid with a triangular base.
    h. Figure A - this is a feature of a prism.


5. Select all units that can be used for surface area. Explain why the others cannot be used for surface area.

A. square meters
B. feet
C. centimeters
D. cubic inches
E. square inches
F. square feet

  • Answers

    A, E, and F are all units that can be used for surface area, as they are square units (two-dimensional).
    B and C cannot be used as they are units of length. D cannot be used as it is a cubic unit, a unit of volume (three-dimensional).


6. Find the area of this polygon. Show your reasoning.

An image of a 7-sided polygon. The bottom side of the polygon is six units long, and extends out to a total width of 10 units, and a central height of 6 units.

  • Answers

    An image of a 7-sided polygon. The bottom side of the polygon is six units long, and extends out to a total width of 10 units, and a central height of 6 units. This polygon has been divided into 3 triangles and 1 rectangle. 2 of the triangles have base 3 and height 2. The third triangle has base 6 and height 3. The rectangle is 6 by 3.

    The polygon can be divided into 3 triangles and 1 rectangle.

    The total area of these shapes, and the area of the polygon, is 2(½ × 2 × 3) + (½ × 6 × 3) + (6 × 3) = 33 square units.



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