Surface Area Nets

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Surface Area Nets · Grade 6 · Worksheet 1

1. A rectangular prism has dimensions of 12 cm by 8 cm by 5 cm. If you were to create a net of this prism by cutting along some edges and flattening it out, what would be the total surface area of all the faces combined? Answer: 392 cm² Solution: A rectangular prism has 6 faces. Each pair of opposite faces are identical rectangles. The dimensions are: Length = 12 cm Width = 8 cm Height = 5 cm Identify the three different face pairs and their areas.
   Full step-by-step solution

   Let's find the total surface area of the rectangular prism step by step. Step 1: Understand the problem. A rectangular prism has 6 faces. Each pair of opposite faces are identical rectangles. The dimensions are: Length = 12 cm Width = 8 cm Height = 5 cm Step 2: Identify the three different face pairs and their areas. Pair 1: Faces with dimensions 12 cm by 8 cm. Area of one such face = 12 × 8 = 96 cm² There are 2 of them, so total = 2 × 96 = 192 cm² Pair 2: Faces with dimensions 12 cm by 5 cm. Area of one such face = 12 × 5 = 60 cm² There are 2 of them, so total = 2 × 60 = 120 cm² Pair 3: Faces with dimensions 8 cm by 5 cm. Area of one such face = 8 × 5 = 40 cm² There are 2 of them, so total = 2 × 40 = 80 cm² Step 3: Add the areas of all faces. Total surface area = 192 + 120 + 80 Step 4: Perform the addition. 192 + 120 = 312 312 + 80 = 392 Step 5: State the final answer with units. Total surface area = 392 cm² This is the area of the net, since a net shows all faces of the prism.

2. A rectangular prism has a net consisting of 6 rectangles. The dimensions of the faces are: two faces measure 12 cm by 8 cm, two faces measure 12 cm by 5 cm, and two faces measure 8 cm by 5 cm. What is the total surface area of this rectangular prism? Answer: 392 cm² Solution: A rectangular prism has 6 faces. - Two faces: 12 cm by 8 cm - Two faces: 12 cm by 5 cm - Two faces: 8 cm by 5 cm We need to find the total surface area. 1.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** A rectangular prism has 6 faces. The net shows: - Two faces: 12 cm by 8 cm - Two faces: 12 cm by 5 cm - Two faces: 8 cm by 5 cm We need to find the total surface area. --- **Step 2: Calculate the area of each type of face** 1. Area of one 12 cm by 8 cm face: Area = length × width = 12 × 8 = 96 cm² There are two such faces: 2 × 96 = 192 cm² 2. Area of one 12 cm by 5 cm face: Area = 12 × 5 = 60 cm² There are two such faces: 2 × 60 = 120 cm² 3. Area of one 8 cm by 5 cm face: Area = 8 × 5 = 40 cm² There are two such faces: 2 × 40 = 80 cm² --- **Step 3: Add them up** Total surface area = 192 + 120 + 80 = 312 + 80 = 392 cm² --- **Step 4: Conclusion** The total surface area is **392 cm²**.

3. Noah is designing a custom pyramid-shaped gift box for his friend's birthday. The box has a square base with side length 12 cm and four triangular faces that are isosceles triangles with a base of 12 cm and a slant height of 10 cm. He needs to calculate the total surface area to determine how much decorative paper he needs to cover all five faces. What is the total surface area of Noah's pyramid box in square centimeters? Answer: 384 Solution: Area of square = side × side = 12 cm × 12 cm = 144 square cm Area of triangle = (1/2) × base × height = (1/2) × 12 cm × 10 cm = 60 square cm Calculate the total area of all four triangular faces Total triangular area = 4 × 60 square cm = 240 square cm Total surface area = base area + triangular…
   Full step-by-step solution

   Step 1: Calculate the area of the square base Area of square = side × side = 12 cm × 12 cm = 144 square cm Step 2: Calculate the area of one triangular face Area of triangle = (1/2) × base × height = (1/2) × 12 cm × 10 cm = 60 square cm Step 3: Calculate the total area of all four triangular faces Total triangular area = 4 × 60 square cm = 240 square cm Step 4: Calculate the total surface area Total surface area = base area + triangular area = 144 square cm + 240 square cm = 384 square cm The answer is 384.

4. A net of a rectangular prism has faces with dimensions: two faces are 25 cm by 20 cm, two faces are 25 cm by 15 cm, and two faces are 20 cm by 15 cm. What is the surface area of the rectangular prism? Answer: 2350 Solution: Identify the three pairs of faces and their dimensions. - Two faces: 25 cm by 20 cm - Two faces: 25 cm by 15 cm - Two faces: 20 cm by 15 cm Calculate the area of each pair.
   Full step-by-step solution

   Step 1: Identify the three pairs of faces and their dimensions. - Two faces: 25 cm by 20 cm - Two faces: 25 cm by 15 cm - Two faces: 20 cm by 15 cm Step 2: Calculate the area of each pair. Area of 25 cm × 20 cm face = 25 × 20 = 500 square cm Since there are two such faces: 2 × 500 = 1000 square cm Area of 25 cm × 15 cm face = 25 × 15 = 375 square cm Two such faces: 2 × 375 = 750 square cm Area of 20 cm × 15 cm face = 20 × 15 = 300 square cm Two such faces: 2 × 300 = 600 square cm Step 3: Add all the areas together. Total surface area = 1000 + 750 + 600 = 2350 square cm The answer is 2350.

5. A net of a rectangular prism has faces with dimensions: 15 cm by 11 cm, 15 cm by 8 cm, 11 cm by 8 cm, each appearing twice. What is the total surface area? Answer: 746 Solution: Identify the three pairs of opposite faces. The dimensions are 15 cm by 11 cm, 15 cm by 8 cm, and 11 cm by 8 cm. Calculate the area of the first pair: 15 × 11 = 165 square cm.
   Full step-by-step solution

   Step 1: Identify the three pairs of opposite faces. The dimensions are 15 cm by 11 cm, 15 cm by 8 cm, and 11 cm by 8 cm. Step 2: Calculate the area of the first pair: 15 × 11 = 165 square cm. Since there are two such faces, total for this pair is 165 × 2 = 330 square cm. Step 3: Calculate the area of the second pair: 15 × 8 = 120 square cm. Two faces give 120 × 2 = 240 square cm. Step 4: Calculate the area of the third pair: 11 × 8 = 88 square cm. Two faces give 88 × 2 = 176 square cm. Step 5: Add all the areas: 330 + 240 + 176 = 746 square cm. The answer is 746.