Volume of Cylinders, Cones, and Spheres
Mathematics -- Understanding and Applying 3D Volume Formulas
Learning Objectives
Identify the parts of cylinders, cones, and spheres (radius, height, diameter) and explain how they relate to volume calculations
State and apply the volume formula for a cylinder: V = pi r^2 h
State and apply the volume formula for a cone: V = (1/3) pi r^2 * h
State and apply the volume formula for a sphere: V = (4/3) pi r^3
Explain the relationship between the volume of a cylinder and a cone with the same base and height
Volume of 3D Shapes: Why It Matters
Volume measures the amount of space inside a three-dimensional object. It tells you how much a container can hold, how much material is needed to fill a shape, or how much space an object takes up.
You already know how to find the volume of rectangular prisms (length times width times height). But the real world is full of shapes that are not rectangular: soda cans, ice cream cones, basketballs, storage tanks, rocket nose cones, and water towers. To work with these shapes, you need three new formulas.
In this lesson, you will learn the volume formulas for three shapes:
1. Cylinder -- shaped like a can or a pipe 2. Cone -- shaped like an ice cream cone or a funnel 3. Sphere -- shaped like a ball
The great news is that these three formulas are closely related to each other. Once you understand the cylinder formula, the cone and sphere formulas build directly on it.
All three volume formulas use pi (approximately 3.14159). This is because cylinders, cones, and spheres all have circular cross-sections. Whenever you see a circle in a 3D shape, pi will be part of the formula. In this lesson, use pi = 3.14 unless your teacher says otherwise.
Part 1: Volume of a Cylinder
A cylinder is a 3D shape with two identical circular bases connected by a curved surface. Think of a soup can, a drinking glass, or a roll of paper towels.
The Key Parts
- Radius (r): The distance from the center of the circular base to its edge
• Height (h): The distance between the two circular bases (how tall the cylinder is)
• Diameter (d): The distance across the full circle through the center (d = 2r)
Building the Formula
Remember that the area of a circle is A = pi * r^2. A cylinder is basically a circle that has been "stacked" or "stretched" upward to a certain height. So the volume is:
V = pi r^2 h
In words: Volume = (area of the circular base) times (the height)
Example 1: Soup Can
A soup can has a radius of 4 cm and a height of 12 cm. Find its volume.
- V = pi r^2 h
• V = 3.14 (4)^2 12
• V = 3.14 16 12
• V = 3.14 * 192
• V = 602.88 cubic centimeters (cm^3)
Example 2: Water Pipe
A cylindrical water pipe has a diameter of 10 inches and is 36 inches long. Find its volume.
- First, find the radius: r = d / 2 = 10 / 2 = 5 inches
• V = pi r^2 h
• V = 3.14 (5)^2 36
• V = 3.14 25 36
• V = 3.14 * 900
• V = 2,826 cubic inches (in^3)
Notice: Volume is always measured in cubic units (cm^3, in^3, ft^3, m^3) because you are measuring three dimensions.
Part 2: Volume of a Cone
A cone is a 3D shape with one circular base that tapers to a single point (called the apex or vertex). Think of an ice cream cone, a traffic cone, or a funnel.
The Key Parts
- Radius (r): The radius of the circular base
• Height (h): The perpendicular distance from the base to the apex (the tip)
Building the Formula
Here is the amazing connection: a cone with the same base and height as a cylinder holds exactly one-third of the volume. If you poured water from a cone into a cylinder with the same radius and height, you would need to fill the cone exactly three times to fill the cylinder.
So the formula is:
V = (1/3) pi r^2 * h
In words: Volume = one-third of the cylinder with the same base and height
Example 3: Ice Cream Cone
An ice cream cone has a radius of 3 cm and a height of 10 cm. Find its volume.
- V = (1/3) pi r^2 * h
• V = (1/3) 3.14 (3)^2 * 10
• V = (1/3) 3.14 9 * 10
• V = (1/3) 3.14 90
• V = (1/3) * 282.6
• V = 94.2 cm^3
Example 4: Sand Pile
A conical pile of sand has a diameter of 8 feet and a height of 5 feet. Find its volume.
- First, find the radius: r = d / 2 = 8 / 2 = 4 feet
• V = (1/3) pi r^2 * h
• V = (1/3) 3.14 (4)^2 * 5
• V = (1/3) 3.14 16 * 5
• V = (1/3) 3.14 80
• V = (1/3) * 251.2
• V = 83.73 ft^3
A cone is exactly one-third the volume of a cylinder with the same radius and height. This is not a coincidence or an approximation. It is a mathematical fact proven by calculus. If you ever forget the cone formula, just remember: start with the cylinder formula (pi r^2 h), then multiply by one-third.
Part 3: Volume of a Sphere
A sphere is a perfectly round 3D shape where every point on the surface is the same distance from the center. Think of a basketball, a globe, a marble, or the Earth.
The Key Parts
- Radius (r): The distance from the center to any point on the surface
• Diameter (d): The distance across the sphere through the center (d = 2r)
Note: A sphere has no height dimension. The only measurement you need is the radius.
The Formula
V = (4/3) pi r^3
Notice the key difference: for a sphere, you cube the radius (r^3 = r r r), not square it. This is because a sphere extends equally in all three dimensions.
Example 5: Basketball
A basketball has a diameter of 9.4 inches. Find its volume.
- First, find the radius: r = d / 2 = 9.4 / 2 = 4.7 inches
• V = (4/3) pi r^3
• V = (4/3) 3.14 (4.7)^3
• V = (4/3) 3.14 103.823
• V = (4/3) * 326.005
• V = 434.67 in^3
Example 6: Marble
A marble has a radius of 0.5 cm. Find its volume.
- V = (4/3) pi r^3
• V = (4/3) 3.14 (0.5)^3
• V = (4/3) 3.14 0.125
• V = (4/3) * 0.3925
• V = 0.52 cm^3
That is a tiny volume, which makes sense for a marble!
Comparing the Three Formulas
Let's put all three formulas side by side and see the pattern:
| Shape | Formula | Key Feature |
|---|---|---|
| Cylinder | V = pi r^2 h | Full circular base times height |
| Cone | V = (1/3) pi r^2 h | One-third of the cylinder |
| Sphere | V = (4/3) pi r^3 | Radius cubed (no height needed) |
Important Patterns to Remember
1. All three formulas use pi because all three shapes involve circles 2. Cylinder and cone both use r^2 and h (they have a circular base and a height) 3. The cone is exactly one-third of the cylinder with the same base and height 4. The sphere only uses radius because it is the same in every direction 5. The sphere uses r^3 (cubed) instead of r^2 (squared)
Common Mistakes to Avoid
- Forgetting to find the radius from the diameter: If a problem gives the diameter, always divide by 2 first
• Squaring vs. cubing: Cylinders and cones use r^2, but spheres use r^3
• Forgetting the fraction: The cone needs (1/3) and the sphere needs (4/3)
• Wrong units: Volume is always in cubic units (cm^3, in^3, ft^3), not square units
Think of the formulas as a family. The cylinder is the "parent" formula: pi r^2 h. The cone is one-third of the parent. The sphere breaks away from the family by using r^3 instead of r^2 h, but still keeps pi and adds its own fraction (4/3). If you can remember the cylinder formula, you can build the other two from it.
Assessment
Which measurement do ALL three volume formulas (cylinder, cone, and sphere) require?
A sphere with a radius of 5 cm has a greater volume than a cylinder with a radius of 5 cm and a height of 5 cm.
A cylindrical water tank has a radius of 3 feet and a height of 8 feet. Its volume is V = 3.14 x {blank1} x {blank2} = {blank3} cubic feet.
A cylindrical can has a diameter of 6 cm and a height of 10 cm. What is its volume? (Use pi = 3.14)
A cone and a cylinder have the same radius (4 cm) and the same height (9 cm). The volume of the cylinder is 452.16 cm^3. What is the volume of the cone?
A traffic cone has a radius of 5 inches and a height of 18 inches. What is its volume? (Use pi = 3.14)
A spherical ball has a radius of 6 cm. What is its volume? (Use pi = 3.14)
A globe has a diameter of 10 inches. Its radius is {blank1} inches. Using the sphere formula, its volume is V = (4/3) x 3.14 x {blank2} = {blank3} cubic inches. (Round to the nearest whole number.)
Match each formula to the correct 3D shape.
A farmer needs to fill a cone-shaped grain hopper (radius 2 m, height 6 m) and a cylindrical grain bin (radius 2 m, height 6 m). How many times would the farmer need to fill the cone to completely fill the cylinder? Explain your reasoning using what you know about the relationship between cone and cylinder volumes.