Nets of Three-Dimensional Figures - Sixth Grade Mathematics
A net is a two-dimensional flat pattern of connected polygons that can be folded along shared edges to assemble a three-dimensional solid figure.
The Concept of Unfolding a Solid
Imagine a cardboard shipping carton. If you carefully slit the seams and unfold the box flat onto the floor, you create a 2D template. Every face of the 3D solid is represented on the flat plane without overlapping.
Unfolding a Cube:
3D Solid (Cube) 2D Foldable Net
+--------+ +---+
/ /| | 1 | <- Top Face
+--------+ | +---+---+---+---+
| | + | 2 | 3 | 4 | 5 | <- 4 Side Faces
| |/ +---+---+---+---+
+--------+ | 6 | <- Bottom Face
+---+
When folded along the internal dashed lines, faces 1 and 6 become the top and bottom bases, while faces 2, 3, 4, and 5 wrap around to form the four lateral walls.
Standard Nets for Fundamental Solids
Recognizing nets involves identifying the base shapes and the appropriate count and placement of lateral faces.
1. Net of a Rectangular Prism (6 Rectangular Faces):
+-------+
| Top |
+-------+-------+-------+-------+
| Left | Front | Right | Back |
+-------+-------+-------+-------+
| Bottom|
+-------+
2. Net of a Triangular Prism (2 Congruent Triangles + 3 Rectangles):
/\
/ \ <- Triangular Base 1
+-----+
| | <- Lateral Face 1
+-----+
| | <- Lateral Face 2
+-----+
| | <- Lateral Face 3
+-----+
\ / <- Triangular Base 2
\/
3. Net of a Square Pyramid (1 Square Base + 4 Triangles):
/\
/ \
+-+----+--+
/| | | |\
/ | |Base| | \
+--+-+----+--+-+
\ /
\/
4. Net of a Cylinder (2 Congruent Circles + 1 Large Rectangle):
( O ) <- Circular Top
+------------+
| | <- Rectangular Curved Body (Width = Circumference = 2*pi*r)
+------------+
( O ) <- Circular Bottom
Determining Valid Nets vs. Invalid Nets
Not every arrangement of six squares forms a valid cube net. To form a valid closed solid:
1. No two faces can occupy the same position when folded (no overlaps).
2. Opposite faces must fold to face each other without colliding.
3. Every edge must meet a corresponding edge of identical length.
Valid Cube Net: Invalid Cube Net (Overlapping Faces):
+---+ +---+---+
| | | A | B |
+---+---+---+ +---+---+
| | | | | C | D |
+---+---+---+ +---+---+
| | | E | F |
+---+ +---+---+
Folds into a cube! Faces overlap; top/bottom remain open!
There are exactly 11 distinct valid hexomino nets that fold into a closed cube.
Using Nets to Calculate Surface Area
The net is the foundational bridge to calculating the total surface area of any 3D solid: Step 1: Draw the flat net of the solid. Step 2: Label the dimensions of every individual face. Step 3: Calculate the area of each polygonal face. Step 4: Sum all individual face areas together.
Exercises
Exercise 1. Identify which solid figure is formed by each net description: (a) A net consisting of 1 central rectangle surrounded by 4 attached triangles; (b) A net consisting of 2 congruent equilateral triangles and 3 congruent rectangles; (c) A net consisting of 2 congruent circles and 1 large rectangle; (d) A net consisting of 6 congruent squares.
Exercise 2. A rectangular prism has dimensions 5 cm by 3 cm by 2 cm. Describe the dimensions of each of the 6 rectangular faces that appear on its flat net.
Exercise 3. A student claims that any combination of 6 joined squares will fold into a cube. Explain why this claim is incorrect, and give a specific counterexample.
Exercise 4. A cylinder has a height of 10 cm and a circular base with radius 3 cm. When unfolded into a net, what are the exact length and width of the unrolled rectangular lateral face? (Use pi approx. 3.14).
Solutions
Solution to Exercise 1. For (a), 1 rectangle plus 4 triangles folding to a single apex forms a rectangular pyramid. For (b), 2 triangles plus 3 rectangles forms a triangular prism. For (c), 2 circles plus 1 rectangle forms a cylinder. For (d), 6 congruent squares form a cube.
Solution to Exercise 2. The 6 rectangular faces come in 3 congruent opposite pairs: Pair 1 (Front and Back) measures 5 cm by 2 cm each; Pair 2 (Top and Bottom) measures 5 cm by 3 cm each; Pair 3 (Left and Right sides) measures 3 cm by 2 cm each.
Solution to Exercise 3. The claim is incorrect because the relative positions of the squares dictate whether they fold into a closed 3D boundary. For example, six squares arranged in a straight 1 by 6 strip will roll into an open sleeve with two overlapping faces and two open ends, failing to enclose a cube. Similarly, a 2 by 3 block of squares will overlap multiple faces upon folding, leaving opposite sides missing.
Solution to Exercise 4. The width of the unrolled rectangle equals the height of the cylinder, which is 10 cm. The length of the unrolled rectangle equals the circumference of the circular base: C = 2 * pi * r = 2 * 3.14 * 3 = 18.84 cm. Therefore, the unrolled rectangular lateral face measures 18.84 cm by 10 cm.