3D Figure Nets

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Answer Key & Explanations

3D Figure Nets · Grade 6 · Worksheet 3

1. (-18) × 3 + 36 ÷ (-4) = ? Answer: -63 Solution: First, perform the multiplication: (-18) × 3 = -54 Next, perform the division: 36 ÷ (-4) = -9 Now add the results: -54 + (-9) = -54 - 9 = -63 The answer is -63.
   Full step-by-step solution

   Step 1: First, perform the multiplication: (-18) × 3 = -54 Step 2: Next, perform the division: 36 ÷ (-4) = -9 Step 3: Now add the results: -54 + (-9) = -54 - 9 = -63 The answer is -63.

2. Liam is designing a gift box for his friend's birthday. The box is a rectangular prism that needs to be 25 cm long, 18 cm wide, and 12 cm high. If he draws a net of this box on a single sheet of paper, what will be the total area of the paper he uses for the net? Answer: 1932 cm² Solution: A net of a rectangular prism is a 2D shape that can be folded to form the 3D box. It consists of all the faces of the prism laid out flat. Identify the faces and their dimensions.
   Full step-by-step solution

   A net of a rectangular prism is a 2D shape that can be folded to form the 3D box. It consists of all the faces of the prism laid out flat. Step 1: Identify the faces and their dimensions. A rectangular prism has 6 faces, but opposite faces are equal in area. The three different pairs of faces have dimensions: - Length and height: 25 cm by 12 cm - Length and width: 25 cm by 18 cm - Width and height: 18 cm by 12 cm Step 2: Calculate the area of each type of face. - Area of one face that is 25 cm by 12 cm = 25 * 12 = 300 cm² - Area of one face that is 25 cm by 18 cm = 25 * 18 = 450 cm² - Area of one face that is 18 cm by 12 cm = 18 * 12 = 216 cm² Step 3: Determine how many of each face type are in the net. Since it's a net for the entire box, it must include all 6 faces. There are 2 faces of each type. - Two faces of 300 cm² each - Two faces of 450 cm² each - Two faces of 216 cm² each Step 4: Calculate the total area of the net. Total area = (2 * 300) + (2 * 450) + (2 * 216) First, calculate each product: 2 * 300 = 600 2 * 450 = 900 2 * 216 = 432 Now add them together: 600 + 900 = 1500 1500 + 432 = 1932 Step 5: State the final answer. The total area of the paper used for the net is 1932 cm².

3. Maria is designing a cylindrical container to hold 3 liters of lemonade for a school fundraiser. The container needs to have a height of 25 cm. She creates a net for the container consisting of two circles for the bases and one rectangle for the side. 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: 3 liters = 3 × 1000 = 3000 cm³ Volume = π × r² × h 3000 = 3.14 × r² × 25 Multiply 3.14 × 25 3.14 × 25 = 78.5 3000 = 78.5 × r² Divide both sides by 78.5 r² = 3000 ÷ 78.5 3000 ÷ 78.5 ≈ 38.2166 r = sqrt(38.2166) ≈ 6.18 The radius of the circular bases is approximately 6.18 cm.
   Full step-by-step solution

   Step 1: Convert liters to cubic centimeters 3 liters = 3 × 1000 = 3000 cm³ Step 2: Write the volume formula for a cylinder Volume = π × r² × h Step 3: Substitute known values into the formula 3000 = 3.14 × r² × 25 Step 4: Multiply 3.14 × 25 3.14 × 25 = 78.5 Step 5: Rewrite the equation 3000 = 78.5 × r² Step 6: Divide both sides by 78.5 r² = 3000 ÷ 78.5 Step 7: Calculate the division 3000 ÷ 78.5 ≈ 38.2166 Step 8: Take the square root r = sqrt(38.2166) ≈ 6.18 The radius of the circular bases is approximately 6.18 cm.

4. 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

5. A net of a rectangular prism has faces with areas 72 cm², 72 cm², 48 cm², 48 cm², 54 cm², and 54 cm². What is the volume of the prism? Answer: 432 Solution: Let the dimensions be length (l), width (w), and height (h). The three distinct face areas are: lw = 72, wh = 48, lh = 54. Multiply all three equations: (lw)(wh)(lh) = 72 × 48 × 54 → l²w²h² = 186624.
   Full step-by-step solution

   Step 1: Let the dimensions be length (l), width (w), and height (h). The three distinct face areas are: lw = 72, wh = 48, lh = 54. Step 2: Multiply all three equations: (lw)(wh)(lh) = 72 × 48 × 54 → l²w²h² = 186624. Step 3: Take the square root: lwh = √186624 = 432. Step 4: The volume is lwh = 432 cm³. The answer is 432.

6. ∛(125) + 4² = ? Answer: 21 Solution: We need to evaluate the expression: cube root of 125 plus 4 squared. That is: ∛(125) + 4². The cube root of 125 is the number that, when multiplied by itself three times, gives 125.
   Full step-by-step solution

   Step 1: Understand the problem We need to evaluate the expression: cube root of 125 plus 4 squared. That is: ∛(125) + 4². Step 2: Evaluate the cube root The cube root of 125 is the number that, when multiplied by itself three times, gives 125. Since 5 × 5 × 5 = 25 × 5 = 125, ∛(125) = 5. Step 3: Evaluate the square 4² means 4 × 4. 4 × 4 = 16. Step 4: Add the results Now we have: 5 + 16 = 21. Step 5: Final answer The result is 21.

7. A net of a rectangular prism has faces with areas 36 cm², 36 cm², 24 cm², 24 cm², 54 cm², and 54 cm². What are the dimensions (length, width, height) of the prism in centimeters? Answer: 6 cm × 4 cm × 9 cm Solution: In a rectangular prism, opposite faces are congruent. Let the dimensions be length (l), width (w), and height (h). We have three area values: 36, 24, and 54.
   Full step-by-step solution

   Step 1: In a rectangular prism, opposite faces are congruent. So the three pairs of face areas are: 36 and 36, 24 and 24, 54 and 54. Step 2: Let the dimensions be length (l), width (w), and height (h). The three face areas are l×w, l×h, and w×h. Step 3: We have three area values: 36, 24, and 54. So l×w = 36, l×h = 24, w×h = 54 (or any pairing). Step 4: From l×w = 36 and l×h = 24, divide the first by the second: (l×w)/(l×h) = 36/24, so w/h = 3/2, so w = (3/2)h. Step 5: Substitute into w×h = 54: (3/2)h × h = 54, so (3/2)h² = 54, so h² = 36, so h = 6 cm. Step 6: Then w = (3/2)×6 = 9 cm, and l = 36/w = 36/9 = 4 cm. Step 7: Check: l×w = 4×9 = 36, l×h = 4×6 = 24, w×h = 9×6 = 54. All match. The dimensions are 4 cm × 9 cm × 6 cm (or any order).