3D Figure Nets

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Answer Key & Explanations

3D Figure Nets · Grade 6 · Worksheet 2

1. Liam is designing a gift box in the shape of a rectangular prism. The net of the box shows that the length is 12 cm, the width is 8 cm, and the height is 5 cm. He needs to calculate the total surface area of the box to determine how much wrapping paper he will need. What is the total surface area of the box in square centimeters? Answer: 392 Solution: Top and bottom faces: Area of one = length × width = 12 × 8 = 96 Two of them: 2 × 96 = 192 2. Front and back faces: Area of one = length × height = 12 × 5 = 60 Two of them: 2 × 60 = 120 3.
   Full step-by-step solution

   Let's find the total surface area of the rectangular prism step by step. --- **Step 1: Understand the formula for surface area of a rectangular prism** A rectangular prism has 6 faces: - Top and bottom (length × width) - Front and back (length × height) - Left and right sides (width × height) The formula is: Surface Area = 2 × (length × width + length × height + width × height) --- **Step 2: Identify the given dimensions** Length = 12 cm Width = 8 cm Height = 5 cm --- **Step 3: Calculate each face area type** 1. Top and bottom faces: Area of one = length × width = 12 × 8 = 96 Two of them: 2 × 96 = 192 2. Front and back faces: Area of one = length × height = 12 × 5 = 60 Two of them: 2 × 60 = 120 3. Left and right faces: Area of one = width × height = 8 × 5 = 40 Two of them: 2 × 40 = 80 --- **Step 4: Add them up** Total surface area = 192 + 120 + 80 = 312 + 80 = 392 --- **Step 5: Conclusion** The total surface area is 392 square centimeters. This is the amount of wrapping paper Liam needs.

2. Liam is designing a gift box for his friend's birthday. He creates a net for a rectangular prism where the length is 12 cm, the width is 8 cm, and the height is 5 cm. What is the total surface area of the gift box that Liam is designing? Answer: 392 cm² Solution: A rectangular prism has 6 faces. Opposite faces are equal. Surface Area = 2 * (length * width + length * height + width * height) Identify the given dimensions.
   Full step-by-step solution

   Let's find the total surface area of the rectangular prism (gift box). Step 1: Understand the formula for surface area of a rectangular prism. A rectangular prism has 6 faces. Opposite faces are equal. The formula is: Surface Area = 2 * (length * width + length * height + width * height) Step 2: Identify the given dimensions. Length (l) = 12 cm Width (w) = 8 cm Height (h) = 5 cm Step 3: Calculate the area of each unique face pair. First pair: length * width Area = 12 * 8 = 96 cm² There are 2 such faces, so total = 2 * 96 = 192 cm² Second pair: length * height Area = 12 * 5 = 60 cm² There are 2 such faces, so total = 2 * 60 = 120 cm² Third pair: width * height Area = 8 * 5 = 40 cm² There are 2 such faces, so total = 2 * 40 = 80 cm² Step 4: Add all the totals from the three pairs. Total Surface Area = 192 + 120 + 80 First, 192 + 120 = 312 Then, 312 + 80 = 392 Step 5: State the final answer with units. The total surface area of the gift box is 392 cm². Alternatively, using the formula directly: Surface Area = 2 * (12*8 + 12*5 + 8*5) = 2 * (96 + 60 + 40) = 2 * (196) = 392 cm² Both methods give the same result: 392 cm².

3. A company is designing a new cereal box in the shape of a rectangular prism. The net of the box shows three different rectangle sizes: two large rectangles measuring 25 cm by 40 cm, two medium rectangles measuring 15 cm by 40 cm, and two small rectangles measuring 15 cm by 25 cm. What is the total surface area of this cereal box in square centimeters? Answer: 3950 Solution: A rectangular prism has 6 faces, each pair of opposite faces being identical in size.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Understand the net of the box** A rectangular prism has 6 faces, each pair of opposite faces being identical in size. From the problem: - Two large rectangles: 25 cm × 40 cm - Two medium rectangles: 15 cm × 40 cm - Two small rectangles: 15 cm × 25 cm This means the dimensions of the box are: Length = 40 cm, Width = 25 cm, Height = 15 cm (or some permutation, but it doesn't matter for surface area). --- **Step 2: Calculate area of each pair of faces** 1. **Large rectangles (25 × 40)** Area of one = 25 × 40 = 1000 cm² Two of them: 2 × 1000 = 2000 cm² 2. **Medium rectangles (15 × 40)** Area of one = 15 × 40 = 600 cm² Two of them: 2 × 600 = 1200 cm² 3. **Small rectangles (15 × 25)** Area of one = 15 × 25 = 375 cm² Two of them: 2 × 375 = 750 cm² --- **Step 3: Add them up** Total surface area = 2000 + 1200 + 750 First: 2000 + 1200 = 3200 Then: 3200 + 750 = 3950 --- **Step 4: Final answer** Total surface area = 3950 cm²