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 it has a length of 12 cm, a width of 8 cm, and a height of 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: A rectangular prism has 6 faces: top, bottom, front, back, left side, right side. 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 step by step. Step 1: Understand the formula for the surface area of a rectangular prism. A rectangular prism has 6 faces: top, bottom, front, back, left side, right side. 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 the area of each type of face pair. First pair: length * width (top and bottom faces) Area = 12 * 8 = 96 There are 2 such faces, so total = 2 * 96 = 192 Second pair: length * height (front and back faces) Area = 12 * 5 = 60 There are 2 such faces, so total = 2 * 60 = 120 Third pair: width * height (left and right side faces) Area = 8 * 5 = 40 There are 2 such faces, so total = 2 * 40 = 80 Step 4: Add all these totals together to get the total surface area. Total Surface Area = 192 + 120 + 80 Total Surface Area = 312 + 80 Total Surface Area = 392 Step 5: State the final answer. The total surface area of the box is 392 square centimeters.

2. A company is designing a new cereal box in the shape of a rectangular prism. The net of the box shows rectangles arranged with dimensions: two rectangles measuring 15 cm by 25 cm, two rectangles measuring 15 cm by 8 cm, and two rectangles measuring 25 cm by 8 cm. What is the total surface area of the cereal box in square centimeters? Answer: 1390 Solution: A rectangular prism has 6 faces, each pair of opposite faces being identical.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the net of the rectangular prism** A rectangular prism has 6 faces, each pair of opposite faces being identical. From the problem: - Two rectangles: 15 cm by 25 cm - Two rectangles: 15 cm by 8 cm - Two rectangles: 25 cm by 8 cm --- **Step 2: Identify the surface area formula** Total surface area = sum of the areas of all 6 faces. Since opposite faces are equal, we can compute: Surface area = 2 × (area of one 15×25 face) + 2 × (area of one 15×8 face) + 2 × (area of one 25×8 face) --- **Step 3: Calculate each face area** Area of 15×25 rectangle = 15 × 25 = 375 cm² Area of 15×8 rectangle = 15 × 8 = 120 cm² Area of 25×8 rectangle = 25 × 8 = 200 cm² --- **Step 4: Multiply by 2 for each pair** Two 15×25 faces: 2 × 375 = 750 cm² Two 15×8 faces: 2 × 120 = 240 cm² Two 25×8 faces: 2 × 200 = 400 cm² --- **Step 5: Add them up** Total surface area = 750 + 240 + 400 = 750 + 240 = 990 = 990 + 400 = 1390 cm² --- **Final Answer:** 1390

3. A rectangular prism has dimensions 8 cm × 5 cm × 12 cm. What is its surface area? Answer: 392 cm² Solution: A rectangular prism has 6 faces. The surface area is the sum of the areas of all 6 faces. Surface Area = 2 * (length * width + length * height + width * height) Identify the dimensions.
   Full step-by-step solution

   Let's find the 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. The surface area is the sum of the areas of all 6 faces. The formula is: Surface Area = 2 * (length * width + length * height + width * height) Step 2: Identify the dimensions. Length (l) = 8 cm Width (w) = 5 cm Height (h) = 12 cm Step 3: Calculate the area of each type of face. There are three pairs of identical faces: - Front and back faces: length × height = 8 × 12 = 96 cm² each - Left and right faces: width × height = 5 × 12 = 60 cm² each - Top and bottom faces: length × width = 8 × 5 = 40 cm² each Step 4: Calculate the total area for each pair. - Front and back total: 2 × 96 = 192 cm² - Left and right total: 2 × 60 = 120 cm² - Top and bottom total: 2 × 40 = 80 cm² Step 5: Add them together to get the total surface area. Total surface area = 192 + 120 + 80 = 312 + 80 = 392 cm² Alternatively, using the formula directly: Surface Area = 2 * (8*5 + 8*12 + 5*12) = 2 * (40 + 96 + 60) = 2 * (196) = 392 cm² Final Answer: 392 cm²