3D Figure Nets

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Answer Key & Explanations

3D Figure Nets · Grade 6 · Worksheet 1

1. Charlotte is designing a net for a rectangular prism gift box. The net is made up of six rectangles. Two rectangles measure 27 cm by 12 cm, two rectangles measure 27 cm by 7 cm, and two rectangles measure 12 cm by 7 cm. Which rectangle in the net forms the top and bottom of the prism? Answer: The 27 cm by 12 cm rectangles form the top and bottom. Solution: Identify the three different rectangle sizes in the net: 27 cm by 12 cm, 27 cm by 7 cm, and 12 cm by 7 cm. Step 2: In a rectangular prism, opposite faces are congruent.
   Full step-by-step solution

   Step 1: Identify the three different rectangle sizes in the net: 27 cm by 12 cm, 27 cm by 7 cm, and 12 cm by 7 cm. Step 2: In a rectangular prism, opposite faces are congruent. So there are three pairs of faces. Step 3: The dimensions of the prism are the three side lengths: 27 cm, 12 cm, and 7 cm. Step 4: The top and bottom faces are the ones with the two largest dimensions, which are 27 cm and 12 cm. So the rectangles measuring 27 cm by 12 cm form the top and bottom. The answer is: The 27 cm by 12 cm rectangles form the top and bottom.

2. Aroha is designing a net for a rectangular prism that will be used as a display box. The net has six rectangular faces. Two rectangles measure 27 cm by 11 cm, two rectangles measure 27 cm by 13 cm, and two rectangles measure 11 cm by 13 cm. When the net is folded into a rectangular prism, which two faces will be opposite each other? Describe the pairs of opposite faces by their dimensions. Answer: The pairs of opposite faces are: two 27 cm by 11 cm rectangles opposite each other, two 27 cm by 13 cm rectangles opposite each other, and two 11 cm by 13 cm rectangles opposite each other. Solution: A rectangular prism has 6 faces. Opposite faces are always identical in size and shape. The net has three pairs of identical rectangles: (27 by 11), (27 by 13), and (11 by 13).
   Full step-by-step solution

   Step 1: A rectangular prism has 6 faces. Opposite faces are always identical in size and shape. The net has three pairs of identical rectangles: (27 by 11), (27 by 13), and (11 by 13). Step 2: In a typical cross-shaped net, there is a central rectangle with four rectangles attached to its four edges, and a sixth rectangle attached to one of those four. When folded, the rectangle attached to the top edge becomes opposite the rectangle attached to the bottom edge. The rectangle attached to the left edge becomes opposite the rectangle attached to the right edge. The central rectangle becomes opposite the sixth rectangle. Step 3: Since opposite faces must be identical, the two 27 by 11 rectangles are opposite each other, the two 27 by 13 rectangles are opposite each other, and the two 11 by 13 rectangles are opposite each other. The answer is: The pairs of opposite faces are two 27 cm by 11 cm rectangles, two 27 cm by 13 cm rectangles, and two 11 cm by 13 cm rectangles.

3. Noah is designing a cylindrical storage container for his art supplies. The container needs to hold exactly 3 liters of paint. He creates a net for the container consisting of two circles for the bases and one rectangle for the side. If the height of the container is 25 cm, what is the radius of the circular bases in centimeters? (Use π ≈ 3.14 and remember that 1 liter = 1000 cubic centimeters) Answer: 6.18 Solution: Convert the volume from liters to cubic centimeters. 3 liters × 1000 cm³/liter = 3000 cm³. Use the formula for the volume of a cylinder: V = πr²h, where V is volume, r is radius, and h is height.
   Full step-by-step solution

   Step 1: Convert the volume from liters to cubic centimeters. 3 liters × 1000 cm³/liter = 3000 cm³. Step 2: Use the formula for the volume of a cylinder: V = πr²h, where V is volume, r is radius, and h is height. Step 3: Substitute the known values into the formula: 3000 = 3.14 × r² × 25. Step 4: Calculate 3.14 × 25 = 78.5. Step 5: The equation becomes: 3000 = 78.5 × r². Step 6: Divide both sides by 78.5 to isolate r²: r² = 3000 ÷ 78.5. Step 7: Calculate 3000 ÷ 78.5 ≈ 38.2166. Step 8: Take the square root of both sides to find r: r = sqrt(38.2166) ≈ 6.18 cm. The radius of the circular bases is approximately 6.18 cm.

4. A net of a rectangular prism has faces with areas 48 cm², 48 cm², 30 cm², 30 cm², 40 cm², and 40 cm². What are the dimensions (length, width, height) of the prism? Answer: 8 cm, 6 cm, 5 cm Solution: In a rectangular prism, opposite faces are congruent. The areas given are 48, 30, and 40. Let the dimensions be l, w, h.
   Full step-by-step solution

   Step 1: In a rectangular prism, opposite faces are congruent. So the three distinct areas are length × width, length × height, and width × height. Step 2: The areas given are 48, 30, and 40. Let the dimensions be l, w, h. Step 3: We have l × w = 48, l × h = 40, w × h = 30. Step 4: Multiply all three equations: (l × w) × (l × h) × (w × h) = 48 × 40 × 30 Step 5: This simplifies to l² × w² × h² = 57600 Step 6: Take the square root: l × w × h = sqrt(57600) = 240 Step 7: Now divide the product by each area to find the missing dimension: l = (l × w × h) ÷ (w × h) = 240 ÷ 30 = 8 w = (l × w × h) ÷ (l × h) = 240 ÷ 40 = 6 h = (l × w × h) ÷ (l × w) = 240 ÷ 48 = 5 Step 8: The dimensions are 8 cm, 6 cm, and 5 cm. The answer is 8 cm, 6 cm, 5 cm.

5. A net of a rectangular prism has faces with areas 144 cm², 144 cm², 96 cm², 96 cm², 216 cm², and 216 cm². What are the dimensions of the prism? Answer: 12 cm × 8 cm × 18 cm Solution: In a rectangular prism, opposite faces have the same area. So we have three different face areas: 144 cm², 96 cm², and 216 cm². Let the dimensions be length (l), width (w), and height (h).
   Full step-by-step solution

   Step 1: In a rectangular prism, opposite faces have the same area. So we have three different face areas: 144 cm², 96 cm², and 216 cm². Step 2: Let the dimensions be length (l), width (w), and height (h). Then: l × w = 144 l × h = 96 w × h = 216 Step 3: Multiply all three equations: (l × w) × (l × h) × (w × h) = 144 × 96 × 216 l² × w² × h² = 144 × 96 × 216 (l × w × h)² = 144 × 96 × 216 Step 4: Compute 144 × 96 = 13824. Then 13824 × 216 = 2985984. So (l × w × h)² = 2985984 l × w × h = √2985984 = 1728 Step 5: Now find each dimension: l = (l × w × h) ÷ (w × h) = 1728 ÷ 216 = 8 cm w = (l × w × h) ÷ (l × h) = 1728 ÷ 96 = 18 cm h = (l × w × h) ÷ (l × w) = 1728 ÷ 144 = 12 cm Step 6: The dimensions are 8 cm × 18 cm × 12 cm (order can vary). The answer is 12 cm × 8 cm × 18 cm.