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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). In Geometry Foundations, the goal is not just to get an answer but to recognize the structure of the problem, choose a reliable strategy, and explain why the result is reasonable. The practice set now includes targeted skill work, transfer questions, and mixed review so students build fluency and retention.

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.
  3. Connect the calculation back to Volume of Cylinders, Cones, and Spheres so the method, not just the arithmetic, is clear.
  4. Check the result against the original question before writing the final answer.

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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