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Volume of Cylinders, Cones, and Spheres

A free Pre-Algebra lesson from the “Geometry Foundations” unit, with a worked example and practice problems including step-by-step solutions.

A cylinder's volume is the base area (pi r^2) times the height. A cone has 1/3 the volume of a cylinder with the same base and height. A sphere's volume is (4/3) pi r^3 — about two-thirds of the cylinder that snugly contains it (Archimedes).

What you'll learn

Why it matters: Soup cans (cylinder), ice cream cones, scoops, balloons and balls (sphere), and water-tank capacity calculations all use these three volume formulas.

Worked example

Problem. A cylinder has r = 3 and h = 10. Find V (pi = 3.14).

  1. V = pi * r^2 * h = 3.14 * 9 * 10.
  2. = 3.14 * 90 = 282.6.

Answer: 282.6

Practice problems

1. Cylinder r = 2, h = 10 (pi = 3.14).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Cylinder r = 2, h = 10 (pi = 3.14).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. V = 3.14 * 4 * 10.
  4. = 125.6.
  5. Check the result by substituting or estimating: the response should match 125.6 and make sense in the original problem.

Answer: 125.6

2. Cylinder r = 5, h = 4 (pi = 3.14).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Cylinder r = 5, h = 4 (pi = 3.14).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. V = 3.14 * 25 * 4.
  4. = 314.
  5. Check the result by substituting or estimating: the response should match 314 and make sense in the original problem.

Answer: 314

3. Cone r = 3, h = 10 (pi = 3.14).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Cone r = 3, h = 10 (pi = 3.14).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. V = (1/3) * 3.14 * 9 * 10.
  4. = (1/3) * 282.6 = 94.2.
  5. Check the result by substituting or estimating: the response should match 94.2 and make sense in the original problem.

Answer: 94.2

4. Cone r = 6, h = 5 (pi = 3.14).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Cone r = 6, h = 5 (pi = 3.14).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. V = (1/3) * 3.14 * 36 * 5.
  4. = (1/3) * 565.2 = 188.4.
  5. Check the result by substituting or estimating: the response should match 188.4 and make sense in the original problem.

Answer: 188.4

5. Sphere r = 3 (pi = 3.14).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Sphere r = 3 (pi = 3.14).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. V = (4/3) * 3.14 * 27.
  4. = 4 * 28.26 = 113.04.
  5. Check the result by substituting or estimating: the response should match 113.04 and make sense in the original problem.

Answer: 113.04

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