3D Figure Nets

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Answer Key & Explanations

3D Figure Nets · Grade 6 · Worksheet 1

1. Emma is designing a cylindrical container to hold 2 liters of juice. The container needs to have a height of 20 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: 5.64 Solution: Convert 2 liters to cubic centimeters: 2 × 1000 = 2000 cm³ Use the volume formula for a cylinder: V = πr²h Substitute known values: 2000 = 3.14 × r² × 20 Calculate 3.14 × 20 = 62.8 Rewrite equation: 2000 = 62.8 × r² Divide both sides by 62.8: r² = 2000 ÷ 62.8 Calculate 2000 ÷ 62.8 ≈ 31.847 Take…
   Full step-by-step solution

   Step 1: Convert 2 liters to cubic centimeters: 2 × 1000 = 2000 cm³ Step 2: Use the volume formula for a cylinder: V = πr²h Step 3: Substitute known values: 2000 = 3.14 × r² × 20 Step 4: Calculate 3.14 × 20 = 62.8 Step 5: Rewrite equation: 2000 = 62.8 × r² Step 6: Divide both sides by 62.8: r² = 2000 ÷ 62.8 Step 7: Calculate 2000 ÷ 62.8 ≈ 31.847 Step 8: Take square root: r ≈ √31.847 ≈ 5.64 Step 9: The radius is approximately 5.64 cm

2. A rectangular prism has dimensions 12 cm × 8 cm × 5 cm. What is its surface area? Answer: 392 cm² Solution: The surface area (SA) is the sum of the areas of all six faces. Formula: SA = 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. The surface area (SA) is the sum of the areas of all six faces. Formula: SA = 2 * (length * width + length * height + width * height) Step 2: Identify the dimensions. Length (l) = 12 cm Width (w) = 8 cm Height (h) = 5 cm Step 3: Calculate the area of the three different types of rectangular faces. Area of face 1 (length × width) = l × w = 12 × 8 = 96 cm² Area of face 2 (length × height) = l × h = 12 × 5 = 60 cm² Area of face 3 (width × height) = w × h = 8 × 5 = 40 cm² Step 4: Apply the surface area formula. SA = 2 × (Area1 + Area2 + Area3) SA = 2 × (96 + 60 + 40) Step 5: Simplify inside the parentheses. 96 + 60 = 156 156 + 40 = 196 Step 6: Multiply by 2. SA = 2 × 196 = 392 Step 7: State the final answer with units. The surface area is 392 cm².

3. In art class, Lucas is making a pyramid decoration. The base of the pyramid is a square with side length 6 cm. The triangular faces each have a base of 6 cm and a height of 10 cm. Lucas needs to draw a net for this pyramid. How many triangles will be in the net besides the square base? Answer: 4 Solution: Recall what a square pyramid looks like. It has one square base and four triangular faces that meet at the top. Step 2: The net must include all faces of the pyramid.
   Full step-by-step solution

   Step 1: Recall what a square pyramid looks like. It has one square base and four triangular faces that meet at the top. Step 2: The net must include all faces of the pyramid. Step 3: The square base is one piece. The triangular faces are separate pieces attached to the sides of the square. Step 4: Count the triangles: there are 4 triangular faces. The answer is 4.

4. A net of a rectangular prism has faces with areas 126 cm², 126 cm², 84 cm², 84 cm², 54 cm², and 54 cm². What are the dimensions of the prism? Answer: 9 cm × 6 cm × 14 cm Solution: In a rectangular prism, opposite faces are equal. Let the dimensions be length (l), width (w), and height (h). The three different face areas are: l×w = 126, l×h = 84, w×h = 54.
   Full step-by-step solution

   Step 1: In a rectangular prism, opposite faces are equal. So the three pairs of face areas are: 126 and 126, 84 and 84, 54 and 54. Step 2: Let the dimensions be length (l), width (w), and height (h). The three different face areas are: l×w = 126, l×h = 84, w×h = 54. Step 3: Multiply all three equations: (l×w) × (l×h) × (w×h) = 126 × 84 × 54 Step 4: This gives l² × w² × h² = 126 × 84 × 54 Step 5: Compute 126 × 84 = 10584, then 10584 × 54 = 571536 Step 6: So (l×w×h)² = 571536. Take square root: l×w×h = √571536 = 756 Step 7: Now divide the product by each face area to find the missing dimension: l = (l×w×h) ÷ (w×h) = 756 ÷ 54 = 14 w = (l×w×h) ÷ (l×h) = 756 ÷ 84 = 9 h = (l×w×h) ÷ (l×w) = 756 ÷ 126 = 6 Step 8: The dimensions are 14 cm × 9 cm × 6 cm (or any order). The answer is 9 cm × 6 cm × 14 cm.

5. (-15) × 4 + 24 ÷ (-8) = ? Answer: -63 Solution: First, perform the multiplication: (-15) × 4 = -60 Next, perform the division: 24 ÷ (-8) = -3 Now add the results: -60 + (-3) = -63 The answer is -63.
   Full step-by-step solution

   Step 1: First, perform the multiplication: (-15) × 4 = -60 Step 2: Next, perform the division: 24 ÷ (-8) = -3 Step 3: Now add the results: -60 + (-3) = -63 The answer is -63.

6. A net of a rectangular prism has faces with areas 35 cm², 35 cm², 21 cm², 21 cm², 15 cm², and 15 cm². What are the dimensions of the prism? Answer: 7 cm × 5 cm × 3 cm Solution: The six faces of a rectangular prism come in three pairs of opposite, equal faces. The areas given are 35, 35, 21, 21, 15, and 15. Let the dimensions be length (l), width (w), and height (h).
   Full step-by-step solution

   Step 1: The six faces of a rectangular prism come in three pairs of opposite, equal faces. The areas given are 35, 35, 21, 21, 15, and 15. So the three distinct areas are 35 cm², 21 cm², and 15 cm². Step 2: Let the dimensions be length (l), width (w), and height (h). The areas are: - l × w = 35 - l × h = 21 - w × h = 15 Step 3: Multiply all three equations together: (l × w) × (l × h) × (w × h) = 35 × 21 × 15 l² × w² × h² = 11025 (l × w × h)² = 11025 l × w × h = sqrt(11025) = 105 Step 4: Now divide the product by each area to find the missing dimension: - h = (l × w × h) ÷ (l × w) = 105 ÷ 35 = 3 cm - w = (l × w × h) ÷ (l × h) = 105 ÷ 21 = 5 cm - l = (l × w × h) ÷ (w × h) = 105 ÷ 15 = 7 cm The dimensions are 7 cm × 5 cm × 3 cm.