Surface Area Nets

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Answer Key & Explanations

Surface Area Nets · Grade 6 · Worksheet 3

1. Liam is building a wooden jewelry box for his mother. The box is shaped like a rectangular prism that is 12 cm long, 8 cm wide, and 5 cm tall. He needs to calculate the total surface area to know how much decorative paper he needs to cover all the outside faces. What is the total surface area of the jewelry box in square centimeters? Answer: 392 Solution: A rectangular prism has 6 faces. Each pair of opposite faces are identical rectangles. Dimensions: length = 12 cm, width = 8 cm, height = 5 cm.
   Full step-by-step solution

   Let's find the total surface area of the rectangular prism jewelry box. Step 1: Understand the problem. A rectangular prism has 6 faces. Each pair of opposite faces are identical rectangles. Dimensions: length = 12 cm, width = 8 cm, height = 5 cm. Step 2: Identify the three pairs of faces. - Top and bottom faces: length × width = 12 × 8 - Front and back faces: length × height = 12 × 5 - Left and right faces: width × height = 8 × 5 Step 3: Calculate the area of each pair. Area of top and bottom = 2 × (12 × 8) = 2 × 96 = 192 cm² Area of front and back = 2 × (12 × 5) = 2 × 60 = 120 cm² Area of left and right = 2 × (8 × 5) = 2 × 40 = 80 cm² Step 4: Add them to get the total surface area. Total surface area = 192 + 120 + 80 = 392 cm² Step 5: Final answer. The total surface area is 392 square centimeters.

2. Mason is designing a custom display case for his collection of model cars. He draws a net of a rectangular prism on a sheet of cardboard. The net shows that the case has a base measuring 37 cm by 22 cm, two side faces measuring 37 cm by 17 cm, and two end faces measuring 22 cm by 17 cm. Mason needs to calculate the total surface area to know how much adhesive film he needs to cover all the exterior surfaces. What is the total surface area of Mason's display case in square centimeters? Answer: 3634 Solution: Identify the three pairs of faces. Top and bottom: each is 37 cm by 22 cm. Front and back: each is 37 cm by 17 cm.
   Full step-by-step solution

   Step 1: Identify the three pairs of faces. Top and bottom: each is 37 cm by 22 cm. Front and back: each is 37 cm by 17 cm. Left and right ends: each is 22 cm by 17 cm. Step 2: Find the area of one top/bottom face. 37 x 22 = 814 square cm. Two faces: 2 x 814 = 1628 square cm. Step 3: Find the area of one front/back face. 37 x 17 = 629 square cm. Two faces: 2 x 629 = 1258 square cm. Step 4: Find the area of one left/right end face. 22 x 17 = 374 square cm. Two faces: 2 x 374 = 748 square cm. Step 5: Add all the areas together. 1628 + 1258 + 748 = 3634 square cm. The total surface area is 3634 square centimeters.

3. Maria is designing a custom pyramid-shaped gift box for her 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. She needs to calculate the total surface area to determine how much decorative paper she needs to cover all five faces. What is the total surface area of Maria's pyramid gift 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 = area of base + area of…
   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 = area of base + area of triangular faces = 144 square cm + 240 square cm = 384 square cm The answer is 384.