3D Volume Formulas

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Answer Key & Explanations

3D Volume Formulas · Grade 8 · Worksheet 1

1. A cylindrical water tank has a radius of 3 meters and a height of 7 meters. What is its volume in cubic meters? (Use π = 3.14) Answer: 197.82 Solution: Recall the formula for the volume of a cylinder. V = π × r² × h where r is the radius and h is the height. Write down the given values.
   Full step-by-step solution

   Step 1: Recall the formula for the volume of a cylinder. The volume V of a cylinder is given by: V = π × r² × h where r is the radius and h is the height. Step 2: Write down the given values. Radius r = 3 meters Height h = 7 meters π = 3.14 Step 3: Substitute the values into the formula. V = 3.14 × (3)² × 7 Step 4: Calculate the square of the radius. (3)² = 3 × 3 = 9 So now: V = 3.14 × 9 × 7 Step 5: Multiply 3.14 by 9. 3.14 × 9 = 28.26 Step 6: Multiply the result by 7. 28.26 × 7 = 197.82 Step 7: State the final answer with units. The volume of the cylindrical water tank is 197.82 cubic meters.

2. Sophia is designing a decorative planter for a garden center. The planter is made of a cylindrical base with a radius of 6 inches and a height of 11 inches. On top of the cylinder sits a conical top (like a lid) with the same radius and a height of 6 inches. What is the total volume of the planter in cubic inches? Use π ≈ 3.14 and round your answer to the nearest whole cubic inch. Answer: 1469 Solution: Find the volume of the cylindrical base. The formula for the volume of a cylinder is V = π * r^2 * h. Here, r = 6 inches and h = 11 inches.
   Full step-by-step solution

   Step 1: Find the volume of the cylindrical base. The formula for the volume of a cylinder is V = π * r^2 * h. Here, r = 6 inches and h = 11 inches. V_cylinder = 3.14 * 6^2 * 11 = 3.14 * 36 * 11 = 3.14 * 396 = 1243.44 cubic inches. Step 2: Find the volume of the conical top. The formula for the volume of a cone is V = (1/3) * π * r^2 * h. Here, r = 6 inches and h = 6 inches. V_cone = (1/3) * 3.14 * 6^2 * 6 = (1/3) * 3.14 * 36 * 6 = (1/3) * 3.14 * 216 = (1/3) * 678.24 = 226.08 cubic inches. Step 3: Add the two volumes together to find the total volume. Total volume = 1243.44 + 226.08 = 1469.52 cubic inches. Rounded to the nearest whole cubic inch, the answer is 1469 cubic inches.

3. Noah is designing a sculpture for a park. The sculpture consists of a cone placed on top of a cylinder. The cylinder has a radius of 7 feet and a height of 10 feet. The cone has the same radius as the cylinder and a height of 9 feet. What is the total volume of the sculpture? (Use π = 3.14) Answer: 2000.96 Solution: Find the volume of the cylinder. The formula is V = π * r^2 * h. Here, r = 7 ft and h = 10 ft.
   Full step-by-step solution

   Step 1: Find the volume of the cylinder. The formula is V = π * r^2 * h. Here, r = 7 ft and h = 10 ft. So, V_cylinder = 3.14 * (7)^2 * 10 = 3.14 * 49 * 10 = 3.14 * 490 = 1538.6 cubic feet. Step 2: Find the volume of the cone. The formula is V = (1/3) * π * r^2 * h. Here, r = 7 ft and h = 9 ft. So, V_cone = (1/3) * 3.14 * (7)^2 * 9 = (1/3) * 3.14 * 49 * 9 = (1/3) * 3.14 * 441 = (1/3) * 1384.74 = 462.36 cubic feet. Step 3: Add the volumes together. Total volume = 1538.6 + 462.36 = 2000.96 cubic feet. Final answer: 2000.96