8.G.C.9 8th Grade Geometry

Volume of Cones, Cylinders, Spheres

Know the formulas for the volumes of cones, cylinders, and spheres, and use them to solve real-world and mathematical problems.

How to explain it

At this standard, students identify and apply the volume formulas for cylinders (V=πr²h), cones (V=⅓πr²h), and spheres (V=4/3πr³) to find volumes in real-world and mathematical contexts.

The anchor students hold onto: Cylinder: V = πr²h. Cone: V = ⅓πr²h (one-third of a cylinder). Sphere: V = 4/3πr³, where r is the radius (cubed for a sphere).

These volume formulas close 8th-grade geometry and carry into high school surface area, composite solids, and applied measurement involving real-world containers, planets, and engineering.

Worked examples

Example 1 Cylinder volume

Cylinder: r = 3 cm, h = 10 cm.

Step 1Substitute into V = πr²h.

Step 2V = π · 3² · 10 = π · 9 · 10.

Step 3V = 90π.

AnswerV = 90π cm³.

Example 2 Cone volume

Cone: r = 6 in, h = 9 in.

Step 1Substitute into V = ⅓πr²h.

Step 2V = ⅓ · π · 6² · 9 = ⅓ · 324π.

Step 3V = 108π.

AnswerV = 108π in³.

Example 3 Sphere volume

Sphere: r = 3 ft; π ≈ 3.14.

Step 1Substitute into V = 4/3πr³.

Step 2V = 4/3 · π · 3³ = 4/3 · 27π = 36π.

Step 3V ≈ 36 · 3.14 ≈ 113.04.

AnswerV ≈ 113 ft³.

Common mistakes

What students write Squaring the radius for a sphere: V = 4/3πr² instead of V = 4/3πr³.

The fix V = 4/3πr³ uses r CUBED — multiply r by itself three times, not twice.

Try this A student found the volume of a sphere with radius 5 cm. Here is their work: V = 4/3 · π · 5² V = 4/3 · 25π V = 100π/3 cm³ Find and explain the error. Then find the correct volume in terms of π.

What students write Forgetting ⅓ on a cone: writing V = πr²h instead of V = ⅓πr²h.

The fix The cone formula is exactly one-third of the same-base cylinder formula.

Teacher tip

Head off the two predictable errors before they happen. First: V = 4/3πr³ uses r CUBED — multiply r by itself three times, not twice. Second: The cone formula is exactly one-third of the same-base cylinder formula.