3D Volume Formulas

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Answer Key & Explanations

3D Volume Formulas · Grade 8 · Worksheet 3

1. Ava is designing a decorative fountain for a park. The fountain consists of a cylindrical basin with a radius of 6 inches and a height of 16 inches. In the center of the basin, there is a spherical ornament with a radius of 6 inches that sits completely submerged in the water. Ava needs to know how much water the fountain can hold when the basin is filled to the brim and the ornament is inside. What is the volume of water the fountain can hold in cubic inches? Use π ≈ 3.14 and round your answer to the nearest whole cubic inch. Answer: 904 Solution: Find the volume of the cylindrical basin. The formula for the volume of a cylinder is V = π × r² × h. Here, r = 6 inches and h = 16 inches.
   Full step-by-step solution

   Step 1: Find the volume of the cylindrical basin. The formula for the volume of a cylinder is V = π × r² × h. Here, r = 6 inches and h = 16 inches. V = 3.14 × 6² × 16 = 3.14 × 36 × 16 = 3.14 × 576 = 1808.64 cubic inches. Step 2: Find the volume of the spherical ornament. The formula for the volume of a sphere is V = (4/3) × π × r³. Here, r = 6 inches. V = (4/3) × 3.14 × 6³ = (4/3) × 3.14 × 216 = (4/3) × 678.24 = 904.32 cubic inches. Step 3: Subtract the volume of the ornament from the volume of the basin to find the water volume. Water volume = 1808.64 - 904.32 = 904.32 cubic inches. Rounded to the nearest whole cubic inch, the answer is 904 cubic inches.

2. Emma is designing a piece of art that consists of a cone placed on top of a cylinder. The cylinder has a radius of 3 inches and a height of 9 inches. The cone has the same radius as the cylinder and a height of 5 inches. What is the total volume of Emma's art piece? Use π = 3.14. Answer: 301.44 Solution: Find the volume of the cylinder. Formula: V_cylinder = π × r^2 × h Given: r = 3 in, h = 9 in, π = 3.14 V_cylinder = 3.14 × (3)^2 × 9 = 3.14 × 9 × 9 = 3.14 × 81 = 254.34 cubic inches Find the volume of the cone.
   Full step-by-step solution

   Step 1: Find the volume of the cylinder. Formula: V_cylinder = π × r^2 × h Given: r = 3 in, h = 9 in, π = 3.14 V_cylinder = 3.14 × (3)^2 × 9 = 3.14 × 9 × 9 = 3.14 × 81 = 254.34 cubic inches Step 2: Find the volume of the cone. Formula: V_cone = (1/3) × π × r^2 × h Given: r = 3 in, h = 5 in, π = 3.14 V_cone = (1/3) × 3.14 × (3)^2 × 5 = (1/3) × 3.14 × 9 × 5 = (1/3) × 3.14 × 45 = (1/3) × 141.3 = 47.1 cubic inches Step 3: Add the volumes together. Total volume = V_cylinder + V_cone = 254.34 + 47.1 = 301.44 cubic inches Final answer: 301.44

3. Emma is creating a decorative candle holder for a craft fair. The holder consists of a cylindrical base with a radius of 3 inches and a height of 7 inches. On top of the cylinder, she places a conical lid with the same radius and a height of 5 inches. Emma wants to know the total volume of the candle holder in cubic inches so she can determine how much wax it will hold. What is the total volume? Use π ≈ 3.14 and round your answer to the nearest whole cubic inch. Answer: 245 Solution: Find the volume of the cylindrical base. Use the formula V = π × r² × h. Here, r = 3 inches and h = 7 inches.
   Full step-by-step solution

   Step 1: Find the volume of the cylindrical base. Use the formula V = π × r² × h. Here, r = 3 inches and h = 7 inches. V = 3.14 × 3² × 7 = 3.14 × 9 × 7 = 3.14 × 63 = 197.82 cubic inches. Step 2: Find the volume of the conical lid. Use the formula V = (1/3) × π × r² × h. Here, r = 3 inches and h = 5 inches. V = (1/3) × 3.14 × 3² × 5 = (1/3) × 3.14 × 9 × 5 = (1/3) × 3.14 × 45 = (1/3) × 141.3 = 47.1 cubic inches. Step 3: Add the volumes together. Total volume = 197.82 + 47.1 = 244.92 cubic inches. Rounded to the nearest whole cubic inch, the answer is 245 cubic inches.