3D Figure Nets

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Answer Key & Explanations

3D Figure Nets · Grade 6 · Worksheet 3

1. Hana is designing a net for a rectangular prism that will be used as a small storage box. The net she draws has six rectangular faces arranged in a cross shape. The dimensions of the faces are: two rectangles measuring 12 cm by 8 cm, two rectangles measuring 12 cm by 6 cm, and two rectangles measuring 8 cm by 6 cm. When folded, which faces will be opposite each other? Describe the pairs of opposite faces by their dimensions. Answer: The pairs of opposite faces are: two 12 cm by 8 cm rectangles opposite each other, two 12 cm by 6 cm rectangles opposite each other, and two 8 cm by 6 cm rectangles opposite each other. Solution: Understand that a rectangular prism has 6 faces, and opposite faces are identical in size. The net shows three pairs of identical rectangles: (12x8), (12x6), and (8x6).
   Full step-by-step solution

   Step 1: Understand that a rectangular prism has 6 faces, and opposite faces are identical in size. The net shows three pairs of identical rectangles: (12x8), (12x6), and (8x6). Step 2: In a typical cross-shaped net, the central rectangle is one face, and the four rectangles around it are attached to its four edges. The sixth rectangle is attached to one of those four. Step 3: When folding, the rectangle attached to the top edge of the central rectangle will become opposite to the rectangle attached to the bottom edge. Similarly, the rectangle attached to the left edge will be opposite to the one attached to the right edge. The central rectangle itself will be opposite to the sixth rectangle that is attached to one of the flaps. Step 4: Since each pair of opposite faces must be identical, we match the sizes: the two 12x8 rectangles are opposite each other, the two 12x6 rectangles are opposite each other, and the two 8x6 rectangles are opposite each other. The answer is: The pairs of opposite faces are two 12 cm by 8 cm rectangles, two 12 cm by 6 cm rectangles, and two 8 cm by 6 cm rectangles.

2. Isabella is designing a net for a square-based pyramid for a school art project. The net must have a square base and four triangular faces. She draws a net with a square in the center measuring 9 cm on each side. She attaches a triangle to each side of the square. Each triangle has a base of 9 cm and a slant height of 14 cm. When Isabella folds this net, will the triangular faces meet at a single point (the apex) above the center of the square base? Explain why or why not using the properties of the net. Answer: Yes, the triangular faces will meet at a single point above the center of the square base because the net has a central square with a triangle attached to each of its four sides, and all four triangles have equal slant heights, allowing them to fold upward and meet at a common apex. Solution: The net has a central square (the base) with side length 9 cm. Attached to each side of the square is an isosceles triangle.
   Full step-by-step solution

   Step 1: Understand the net. The net has a central square (the base) with side length 9 cm. Attached to each side of the square is an isosceles triangle. The base of each triangle matches the side of the square (9 cm), and the slant height (the distance from the apex to the midpoint of the base) is 14 cm for each triangle. Step 2: Visualize folding. When we fold the triangles upward along the edges of the square, each triangle rotates around its base. Since all four triangles are attached to the square (one on each side), they will all rise upward from the base. Step 3: Check if they meet at a single point. For the triangular faces to meet at a single apex, the slant heights must be equal (which they are: all 14 cm), and the triangles must be arranged so that their apexes come together. In this net, each triangle's apex is at the tip opposite its base. When folded, the apex of each triangle will move toward the center of the square. Because the triangles are identical and symmetrically placed around the square, their apexes will all meet at the same point directly above the center of the square base. Step 4: Conclusion. Yes, the triangular faces will meet at a single point (the apex) above the center of the square base. The net is a valid net for a square-based pyramid because it has a central square base with four identical triangular faces attached to each side, and all triangles have the same slant height, allowing them to fold and meet at a common point.

3. A company is designing a new cereal box in the shape of a rectangular prism. The box needs to have a volume of 4320 cubic centimeters. If the length is 18 cm and the width is 12 cm, what is the height of the box? Answer: 20 Solution: We are given a rectangular prism (cereal box) with: - Volume = 4320 cubic cm - Length = 18 cm - Width = 12 cm - Height = unknown, let's call it h. Recall the volume formula for a rectangular prism.
   Full step-by-step solution

   We are given a rectangular prism (cereal box) with: - Volume = 4320 cubic cm - Length = 18 cm - Width = 12 cm - Height = unknown, let's call it h. Step 1: Recall the volume formula for a rectangular prism. Volume = length × width × height Step 2: Substitute the known values into the formula. 4320 = 18 × 12 × h Step 3: Multiply 18 and 12. 18 × 12 = 216 So the equation becomes: 4320 = 216 × h Step 4: Solve for h by dividing both sides by 216. h = 4320 / 216 Step 5: Perform the division. First, note that 216 × 20 = 4320, so: h = 20 Step 6: State the answer with units. The height of the box is 20 cm. Final answer: 20