3D Volume and Surface

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Answer Key & Explanations

3D Volume and Surface · Grade 7 · Worksheet 1

1. Hana is designing a rectangular prism-shaped aquarium with a length of 24 cm, a width of 18 cm, and a height of 12 cm. She wants to cover all six faces with decorative film, then fill the aquarium completely with water. What is the total surface area of the aquarium in square centimeters, and what is its volume in cubic centimeters? Provide your answer as two numbers separated by a comma (e.g., surface area, volume). Answer: 1872, 5184 Solution: Identify the dimensions. Length = 24 cm, Width = 18 cm, Height = 12 cm. Calculate the volume.
   Full step-by-step solution

   Step 1: Identify the dimensions. Length = 24 cm, Width = 18 cm, Height = 12 cm. Step 2: Calculate the volume. Volume = length × width × height = 24 × 18 × 12. First, 24 × 18 = 432. Then, 432 × 12 = 5184 cubic cm. Step 3: Calculate the surface area. Surface area = 2 × (length × width + width × height + height × length). First, compute each area: length × width = 24 × 18 = 432. width × height = 18 × 12 = 216. height × length = 12 × 24 = 288. Sum these: 432 + 216 + 288 = 936. Multiply by 2: 936 × 2 = 1872 square cm. Final answer: Surface area = 1872 square cm, Volume = 5184 cubic cm. So answer is 1872, 5184.

2. Charlotte is designing a rectangular prism-shaped aquarium. The aquarium has a length of 9 cm, a width of 8 cm, and a height of 12 cm. She wants to find both the volume of the aquarium (to know how much water it can hold) and the total surface area (to know how much glass is needed to build it, including the top). What are the volume and total surface area of the aquarium? Provide your answer as two numbers separated by a comma: volume, surface area. Answer: 864, 552 Solution: Find the volume. Volume = length × width × height = 9 × 8 × 12. First, 9 × 8 = 72.
   Full step-by-step solution

   Step 1: Find the volume. Volume = length × width × height = 9 × 8 × 12. First, 9 × 8 = 72. Then, 72 × 12 = 864. So volume = 864 cubic cm. Step 2: Find the surface area. The prism has 6 faces: front/back, left/right, top/bottom. Front and back each have area = 9 × 12 = 108. So total front/back = 2 × 108 = 216. Left and right each have area = 8 × 12 = 96. So total left/right = 2 × 96 = 192. Top and bottom each have area = 9 × 8 = 72. So total top/bottom = 2 × 72 = 144. Total surface area = 216 + 192 + 144 = 552 square cm. Final answer: 864, 552

3. Liam has a rectangular prism-shaped fish tank with a length of 25 cm, a width of 15 cm, and a height of 20 cm. He wants to know how much water it can hold (the volume) and also needs to buy a decorative film to cover all six faces (the surface area). What are the volume and total surface area of the fish tank? Answer: Volume: 7500 cubic cm; Surface Area: 2350 square cm Solution: Identify the dimensions. Length = 25 cm, Width = 15 cm, Height = 20 cm. Calculate the volume.
   Full step-by-step solution

   Step 1: Identify the dimensions. Length = 25 cm, Width = 15 cm, Height = 20 cm. Step 2: Calculate the volume. Volume of a rectangular prism = length × width × height. Volume = 25 × 15 × 20 First, 25 × 15 = 375. Then, 375 × 20 = 7500. So, Volume = 7500 cubic cm. Step 3: Calculate the surface area. A rectangular prism has 6 faces: front/back, top/bottom, left/right. - Front and back faces: each is length × height = 25 × 20 = 500 square cm. Two faces = 2 × 500 = 1000 square cm. - Top and bottom faces: each is length × width = 25 × 15 = 375 square cm. Two faces = 2 × 375 = 750 square cm. - Left and right faces: each is width × height = 15 × 20 = 300 square cm. Two faces = 2 × 300 = 600 square cm. Step 4: Add the areas of all faces. Total Surface Area = 1000 + 750 + 600 = 2350 square cm. Final Answer: Volume = 7500 cubic cm, Surface Area = 2350 square cm.