3D Volume Formulas

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3D Volume Formulas · Grade 8 · Worksheet 3

1. Matiu has a cone-shaped container with a radius of 8 cm and a height of 18 cm. What is its volume in cubic centimeters? (Use π = 3.14) Answer: 1205.76 Solution: Write the formula for the volume of a cone: V = (1/3) × π × r² × h Substitute the given values: r = 8 cm, h = 18 cm, π = 3.14 Calculate r²: 8² = 64 Multiply by π: 3.14 × 64 = 200.96 Multiply by the height: 200.96 × 18 = 3617.28 Multiply by 1/3: 3617.28 ÷ 3 = 1205.76 The answer is 1205.76 cubic…
   Full step-by-step solution

   Step 1: Write the formula for the volume of a cone: V = (1/3) × π × r² × h Step 2: Substitute the given values: r = 8 cm, h = 18 cm, π = 3.14 Step 3: Calculate r²: 8² = 64 Step 4: Multiply by π: 3.14 × 64 = 200.96 Step 5: Multiply by the height: 200.96 × 18 = 3617.28 Step 6: Multiply by 1/3: 3617.28 ÷ 3 = 1205.76 The answer is 1205.76 cubic centimeters.

2. 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.

3. A spherical water storage tank has a diameter of 12 meters. A conical funnel has a height of 4 meters and a base radius of 3 meters. If the spherical tank is completely full and all its water is poured into the conical funnel, how many times will the funnel need to be filled and emptied to transfer all the water? Use π = 3.14 and round your answer to the nearest whole number. Answer: 24 Solution: Radius of sphere = diameter/2 = 12/2 = 6 meters Volume of sphere = (4/3)πr³ = (4/3) × 3.14 × 6³ = (4/3) × 3.14 × 216 = (4/3) × 678.24 = 4 × 226.08 = 904.32 cubic meters Volume of cone = (1/3)πr²h = (1/3) × 3.14 × 3² × 4 = (1/3) × 3.14 × 9 × 4 = (1/3) × 3.14 × 36 = (1/3) × 113.04 = 37.68 cubic…
   Full step-by-step solution

   Step 1: Calculate the volume of the spherical tank Radius of sphere = diameter/2 = 12/2 = 6 meters Volume of sphere = (4/3)πr³ = (4/3) × 3.14 × 6³ = (4/3) × 3.14 × 216 = (4/3) × 678.24 = 4 × 226.08 = 904.32 cubic meters Step 2: Calculate the volume of the conical funnel Volume of cone = (1/3)πr²h = (1/3) × 3.14 × 3² × 4 = (1/3) × 3.14 × 9 × 4 = (1/3) × 3.14 × 36 = (1/3) × 113.04 = 37.68 cubic meters Step 3: Calculate how many times the funnel needs to be filled Number of fills = volume of sphere ÷ volume of cone = 904.32 ÷ 37.68 = 24 The answer is 24.

4. Matiu is designing a spherical sculpture for the town square. He plans to place it on top of a cylindrical pedestal. The sphere has a radius of 21 inches. The cylindrical pedestal has a radius of 21 inches and a height of 42 inches. Matiu wants to paint the entire outer surface of both the sphere and the cylinder, but first he needs to know the total volume of the combined sculpture and pedestal. What is the total volume in cubic inches? (Use π ≈ 22/7) Answer: 97020 Solution: Find the volume of the sphere. Radius r = 21 inches. Volume of sphere = 4/3 × π × r^3 = 4/3 × (22/7) × 21^3 First, calculate 21^3 = 21 × 21 × 21 = 441 × 21 = 9261.
   Full step-by-step solution

   Step 1: Find the volume of the sphere. Radius r = 21 inches. Volume of sphere = 4/3 × π × r^3 = 4/3 × (22/7) × 21^3 First, calculate 21^3 = 21 × 21 × 21 = 441 × 21 = 9261. Then, 4/3 × (22/7) × 9261 = (4 × 22 × 9261) / (3 × 7) = (88 × 9261) / 21 = 814968 / 21 = 38808 cubic inches. Step 2: Find the volume of the cylinder. Radius r = 21 inches, height h = 42 inches. Volume of cylinder = π × r^2 × h = (22/7) × 21^2 × 42 First, calculate 21^2 = 441. Then, (22/7) × 441 × 42 = 22 × (441/7) × 42 = 22 × 63 × 42 = 22 × 2646 = 58212 cubic inches. Step 3: Add the volumes together. Total volume = sphere volume + cylinder volume = 38808 + 58212 = 97020 cubic inches. The total volume is 97020 cubic inches.

5. A spherical water tank has a radius of 9 feet. What is its volume in cubic feet? (Use π = 3.14) Answer: 3052.08 Solution: Use the formula for the volume of a sphere: V = (4/3)πr³ Substitute the given radius: r = 9 feet, π = 3.14 Calculate r³: 9³ = 9 × 9 × 9 = 81 × 9 = 729 Multiply by π: 729 × 3.14 = 2289.06 Multiply by 4/3: (4/3) × 2289.06 = (4 × 2289.06) ÷ 3 = 9156.24 ÷ 3 = 3052.08 The volume of the spherical…
   Full step-by-step solution

   Step 1: Use the formula for the volume of a sphere: V = (4/3)πr³ Step 2: Substitute the given radius: r = 9 feet, π = 3.14 Step 3: Calculate r³: 9³ = 9 × 9 × 9 = 81 × 9 = 729 Step 4: Multiply by π: 729 × 3.14 = 2289.06 Step 5: Multiply by 4/3: (4/3) × 2289.06 = (4 × 2289.06) ÷ 3 = 9156.24 ÷ 3 = 3052.08 The volume of the spherical water tank is 3052.08 cubic feet.

6. π × (5²) × 12 = ? Answer: 300π Solution: We are given: π × (5²) × 12 Calculate 5² 5² means 5 × 5 = 25. So now we have: π × 25 × 12 Multiply 25 and 12 25 × 12 = 25 × (10 + 2) = 25×10 + 25×2 = 250 + 50 = 300. So now we have: π × 300 π × 300 is the same as 300π.
   Full step-by-step solution

   Let's solve this step by step. We are given: π × (5²) × 12 Step 1: Calculate 5² 5² means 5 × 5 = 25. So now we have: π × 25 × 12 Step 2: Multiply 25 and 12 25 × 12 = 25 × (10 + 2) = 25×10 + 25×2 = 250 + 50 = 300. So now we have: π × 300 Step 3: Write the final expression π × 300 is the same as 300π. Final answer: 300π