Volume of Cones, Cylinders, and Spheres
Grade 8 · Math · Study Guide
Calculate the volume of three important 3D shapes using pi and radius measurements.
Volume measures how much 3-dimensional space an object takes up. For curved 3D shapes, we use formulas involving pi (approximately 3.14159) and the radius. The three key shapes in 8th grade are cylinders, cones, and spheres.
A cylinder is like a stack of circles — its volume is the area of its circular base times its height: V = pi x r^2 x h. A cone is one-third of a cylinder with the same base and height: V = (1/3) x pi x r^2 x h. A sphere is a perfectly round ball: V = (4/3) x pi x r^3. A cone is exactly 1/3 the volume of the cylinder that contains it.
Always include cubic units in your final answer (cm^3, in^3, m^3). Real-world uses include figuring out how much water fits in a tank, the volume of an ice cream cone, or the amount of material in a ball.
| Shape | Formula | Key Relationship |
|---|---|---|
| Cylinder | V = pi x r^2 x h | Base area times height |
| Cone | V = (1/3) x pi x r^2 x h | One-third of the enclosing cylinder |
| Sphere | V = (4/3) x pi x r^3 | Depends only on radius |
Key terms
- Volume
- The amount of 3D space an object occupies; measured in cubic units.
- Radius (r)
- The distance from the center to the edge of a circle or sphere.
- Height (h)
- The perpendicular distance between the bases of a cylinder or apex to base of a cone.
- Pi
- Approximately 3.14159; the ratio of circumference to diameter.
- Cylinder
- A 3D shape with two parallel circular bases connected by a curved surface.
- Cone
- A 3D shape with a circular base tapering to a single point (apex).
Flashcards
- Volume of a cylinder
- V = pi x r^2 x h — area of circular base times height.
- Volume of a cone
- V = (1/3) x pi x r^2 x h — one-third of the matching cylinder.
- Volume of a sphere
- V = (4/3) x pi x r^3 — depends only on the radius, cubed.
- Cone vs. cylinder relationship
- V_cone = (1/3) x V_cylinder when they share the same radius and height.
- Cubic units
- Volume units: cm^3, in^3, m^3 — always cubed.
- Radius from diameter
- r = d divided by 2; always halve the diameter before using volume formulas.
Practice questions
What is the volume of a cylinder with radius 3 cm and height 5 cm? (Use pi = 3.14)
- 47.1 cm^3
- 94.2 cm^3
- 141.3 cm^3
- 188.4 cm^3
Show answer
Answer: 141.3 cm^3
V = pi x 9 x 5 = 45 x 3.14 = 141.3 cm^3.
A cone with the same radius and height as a cylinder has exactly half the cylinder's volume.
Show answer
Answer: False
A cone is 1/3 (not 1/2) of the cylinder with the same base and height.
The volume formula for a sphere is V = ___ x pi x r^3.
Show answer
Answer: 4/3
The sphere formula is V = (4/3) x pi x r^3.
A sphere has a radius of 3 in. What is its volume? (Use pi = 3.14, round to nearest tenth)
- 28.3 in^3
- 56.5 in^3
- 113.1 in^3
- 339.3 in^3
Show answer
Answer: 113.1 in^3
V = (4/3) x 3.14 x 27 = (4/3) x 84.78 = 113.1 in^3.
A cone has radius 6 cm and height 9 cm. What is its volume? (Use pi = 3.14)
- 56.5 cm^3
- 339.1 cm^3
- 678.2 cm^3
- 1017.4 cm^3
Show answer
Answer: 339.1 cm^3
V = (1/3) x 3.14 x 36 x 9 = (1/3) x 1017.36 = 339.1 cm^3.
Fun facts
- Archimedes proved that a sphere fits inside a cylinder such that the sphere volume is exactly 2/3 of the cylinder. He was so proud of this that he requested a sphere-in-cylinder engraved on his tombstone.
- A standard ice cream scoop is about 2 inches in diameter. Each scoop is roughly 4.2 cubic inches.
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