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Volume of Cylinders, Cones, and Spheres

A free Geometry lesson from the “Measurement, Circles, and 3D Solids” unit, with a worked example and practice problems including step-by-step solutions.

Volume measures how much space a 3D shape encloses. A cylinder's volume equals the base area (pi * r^2) times the height. A cone with the same base and height has exactly one-third the cylinder's volume. A sphere fits the formula V = (4/3) * pi * r^3 — about two-thirds the volume of the cylinder that snugly contains it (Archimedes).

What you'll learn

Why it matters: Engineering uses these formulas constantly: tank capacities (cylinders), fuel/grain piles (cones), and storage balls or domes (spheres). They are also baseline HSG-GMD.A standards.

Worked example

Problem. Find the volume of a cylinder with r = 4 and h = 7 (use pi = 3.14).

  1. V = pi * r^2 * h = 3.14 * 16 * 7.
  2. = 3.14 * 112 = 351.68.

Answer: 351.68

Practice problems

1. Cylinder r = 3, h = 10 (pi = 3.14).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Cylinder 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 = 3.14 * 9 * 10 = 282.6.
  4. Check the result by substituting or estimating: the response should match 282.6 and make sense in the original problem.

Answer: 282.6

2. Cylinder r = 5, h = 8 (pi = 3.14).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Cylinder r = 5, h = 8 (pi = 3.14).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. V = 3.14 * 25 * 8 = 628.
  4. Check the result by substituting or estimating: the response should match 628 and make sense in the original problem.

Answer: 628

3. Cone r = 6, h = 10 (pi = 3.14).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Cone r = 6, 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 * 36 * 10.
  4. = (1/3) * 1130.4 = 376.8.
  5. Check the result by substituting or estimating: the response should match 376.8 and make sense in the original problem.

Answer: 376.8

4. Cone r = 3, h = 4 (pi = 3.14).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Cone r = 3, h = 4 (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 * 4.
  4. = (1/3) * 113.04 = 37.68.
  5. Check the result by substituting or estimating: the response should match 37.68 and make sense in the original problem.

Answer: 37.68

5. Sphere r = 6 (pi = 3.14).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Sphere r = 6 (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 * 216.
  4. = 904.32.
  5. Check the result by substituting or estimating: the response should match 904.32 and make sense in the original problem.

Answer: 904.32

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