3D Volume and Surface

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Answer Key & Explanations

3D Volume and Surface · Grade 7 · Worksheet 2

1. Mere is building a cylindrical compost bin for her garden. The bin has a radius of 35 cm and a height of 90 cm. She needs to know how much compost the bin can hold (the volume) and how much material she needs to cover the entire outside surface of the bin (the surface area). Find the volume and the total surface area of the cylindrical bin. Use π ≈ 3.14 and round your final answers to the nearest whole number. Answer: Volume = 346,185 cm³, Surface Area = 27,475 cm² Solution: Write down the formulas. Volume of a cylinder: V = π × r² × h Surface area of a cylinder (including top and bottom): SA = 2πrh + 2πr² Identify the given values. r = 35 cm, h = 90 cm, π ≈ 3.14 Calculate the volume.
   Full step-by-step solution

   Step 1: Write down the formulas. Volume of a cylinder: V = π × r² × h Surface area of a cylinder (including top and bottom): SA = 2πrh + 2πr² Step 2: Identify the given values. r = 35 cm, h = 90 cm, π ≈ 3.14 Step 3: Calculate the volume. V = 3.14 × (35)² × 90 First, square the radius: 35² = 35 × 35 = 1225 Then multiply by π: 3.14 × 1225 = 3.14 × 1200 + 3.14 × 25 = 3768 + 78.5 = 3846.5 Then multiply by the height: 3846.5 × 90 = 3846.5 × 9 × 10 = 34618.5 × 10 = 346185 So the volume is 346,185 cm³. Step 4: Calculate the surface area. First, find the lateral surface area (2πrh): 2 × 3.14 × 35 × 90 = 2 × 3.14 × 3150 = 6.28 × 3150 = 6.28 × 3000 + 6.28 × 150 = 18840 + 942 = 19782 So the lateral surface area is 19,782 cm². Next, find the area of the two circular ends (2πr²): 2 × 3.14 × (35)² = 2 × 3.14 × 1225 = 6.28 × 1225 = 6.28 × 1200 + 6.28 × 25 = 7536 + 157 = 7693 So the area of the two ends is 7,693 cm². Add them together: 19782 + 7693 = 27475 So the total surface area is 27,475 cm². Step 5: Round to the nearest whole number (both are already whole numbers). Final answer: Volume = 346,185 cm³, Surface Area = 27,475 cm²

2. A rectangular prism has dimensions 33 cm × 21 cm × 15 cm. Calculate its volume and total surface area. (Format: Volume = ? cm³, Surface Area = ? cm²) Answer: Volume = 10395 cm³, Surface Area = 2646 cm² Solution: Calculate the volume using V = l × w × h V = 33 × 21 × 15 V = 33 × 315 V = 10395 cm³ Calculate the surface area using SA = 2(lw + lh + wh) First, lw = 33 × 21 = 693 cm² Next, lh = 33 × 15 = 495 cm² Then, wh = 21 × 15 = 315 cm² Now add: 693 + 495 + 315 = 1503 cm² Finally, multiply by 2: 2 × 1503…
   Full step-by-step solution

   Step 1: Calculate the volume using V = l × w × h V = 33 × 21 × 15 V = 33 × 315 V = 10395 cm³ Step 2: Calculate the surface area using SA = 2(lw + lh + wh) First, lw = 33 × 21 = 693 cm² Next, lh = 33 × 15 = 495 cm² Then, wh = 21 × 15 = 315 cm² Now add: 693 + 495 + 315 = 1503 cm² Finally, multiply by 2: 2 × 1503 = 3006 cm² The volume is 10395 cm³ and the surface area is 3006 cm².

3. A rectangular prism has dimensions 15 cm × 10 cm × 20 cm. Calculate its volume and total surface area. (Format: Volume, Surface Area) Answer: 3000, 1300 Solution: Calculate the volume using the formula: Volume = length × width × height Volume = 15 cm × 10 cm × 20 cm Volume = 3000 cm³ Calculate the surface area using the formula: Surface Area = 2(lw + lh + wh) First, calculate lw = 15 × 10 = 150 cm² Next, calculate lh = 15 × 20 = 300 cm² Then, calculate wh…
   Full step-by-step solution

   Step 1: Calculate the volume using the formula: Volume = length × width × height Volume = 15 cm × 10 cm × 20 cm Volume = 3000 cm³ Step 2: Calculate the surface area using the formula: Surface Area = 2(lw + lh + wh) First, calculate lw = 15 × 10 = 150 cm² Next, calculate lh = 15 × 20 = 300 cm² Then, calculate wh = 10 × 20 = 200 cm² Now add these three results: 150 + 300 + 200 = 650 cm² Finally, multiply by 2: 2 × 650 = 1300 cm² The volume is 3000 cm³ and the surface area is 1300 cm².