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3D Volume Formulas

Grade 8 · Geometry · Worksheet 1

  1. Aroha is designing a piece of art for a community center. She has a cone-shaped paperweight with a radius of 5 cm and a height of 15 cm. She also has a cylinder-shaped vase with a radius of 5 cm and a height of 15 cm. If she fills both objects completely with colored sand, how much more sand (in cubic centimeters) will the vase hold than the paperweight? (Use π = 3.14)
    Answer: ______________
  2. π × 6² × 15 = ? Answer: ______________
  3. 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: ______________
  4. A cone has a radius of 7 cm and a height of 15 cm. What is its volume in cubic centimeters? (Use π = 3.14) Answer: ______________
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Answer Key & Explanations

3D Volume Formulas · Grade 8 · Worksheet 1

  1. Aroha is designing a piece of art for a community center. She has a cone-shaped paperweight with a radius of 5 cm and a height of 15 cm. She also has a cylinder-shaped vase with a radius of 5 cm and a height of 15 cm. If she fills both objects completely with colored sand, how much more sand (in cubic centimeters) will the vase hold than the paperweight? (Use π = 3.14) Answer: 785 Solution: Find the volume of the cone (paperweight). Volume of cone = (1/3) * pi * r^2 * h r = 5 cm, h = 15 cm, pi = 3.14 Volume = (1/3) * 3.14 * (5)^2 * 15 Volume = (1/3) * 3.14 * 25 * 15 Volume = (1/3) * 3.14 * 375 Volume = 3.14 * 125 Volume = 392.5 cubic cm Find the volume of the cylinder (vase).
    Full step-by-step solution

    Step 1: Find the volume of the cone (paperweight). Volume of cone = (1/3) * pi * r^2 * h r = 5 cm, h = 15 cm, pi = 3.14 Volume = (1/3) * 3.14 * (5)^2 * 15 Volume = (1/3) * 3.14 * 25 * 15 Volume = (1/3) * 3.14 * 375 Volume = 3.14 * 125 Volume = 392.5 cubic cm Step 2: Find the volume of the cylinder (vase). Volume of cylinder = pi * r^2 * h r = 5 cm, h = 15 cm, pi = 3.14 Volume = 3.14 * (5)^2 * 15 Volume = 3.14 * 25 * 15 Volume = 3.14 * 375 Volume = 1177.5 cubic cm Step 3: Find the difference. Difference = Volume of cylinder - Volume of cone Difference = 1177.5 - 392.5 Difference = 785 cubic cm The vase will hold 785 cubic cm more sand than the paperweight. Final Answer: 785

  2. π × 6² × 15 = ? Answer: 540π Solution: We are given: π × 6² × 15 First, calculate 6². 6² means 6 × 6 = 36. 36 × 15 can be broken down as: 36 × 10 = 360 36 × 5 = 180 Now add: 360 + 180 = 540 So 36 × 15 = 540.
    Full step-by-step solution

    Let's solve step by step. We are given: π × 6² × 15 Step 1: First, calculate 6². 6² means 6 × 6 = 36. So the expression becomes: π × 36 × 15 Step 2: Multiply 36 and 15. 36 × 15 can be broken down as: 36 × 10 = 360 36 × 5 = 180 Now add: 360 + 180 = 540 So 36 × 15 = 540. Step 3: Now we have π × 540. This is written as 540π. Final Answer: 540π

  3. 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.

  4. A cone has a radius of 7 cm and a height of 15 cm. What is its volume in cubic centimeters? (Use π = 3.14) Answer: 769.3 Solution: Write the formula for the volume of a cone: V = (1/3) × π × r² × h Substitute the given values: r = 7, h = 15, π = 3.14 V = (1/3) × 3.14 × (7²) × 15 Calculate the square of the radius: 7² = 49 V = (1/3) × 3.14 × 49 × 15 Multiply 49 by 15: 49 × 15 = 735 V = (1/3) × 3.14 × 735 Multiply 3.14 by…
    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 = 7, h = 15, π = 3.14 V = (1/3) × 3.14 × (7²) × 15 Step 3: Calculate the square of the radius: 7² = 49 V = (1/3) × 3.14 × 49 × 15 Step 4: Multiply 49 by 15: 49 × 15 = 735 V = (1/3) × 3.14 × 735 Step 5: Multiply 3.14 by 735: 3.14 × 735 = 2307.9 V = (1/3) × 2307.9 Step 6: Divide by 3: 2307.9 ÷ 3 = 769.3 The volume is 769.3 cubic centimeters.