A cylinder is a circle pushed upwards: volume = πr² × h, the base area times the height.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
A cone is a third of its cylinder
Same base and same height, but one third the volume: πr²h ÷ 3.
The sphere
Volume = 4/3 × πr³. The radius appears three times, because volume is three-dimensional.
Always the radius
All three use the radius. Given a diameter, halve it before anything else.
Three formulas, all built on πr²
Cylinder: πr²h. Cone: one third of πr²h. Sphere: four thirds of πr³. All three start from the circle's area, which is why that formula is worth knowing cold.
Why the cone is a third
Fill a cone with water and pour it into a cylinder of the same base and height. It takes exactly three cones to fill the cylinder. The fraction can be proved with calculus, but it can be demonstrated with sand in a minute.
The sphere uses r cubed
The sphere formula has r³, not r², because a sphere's size is set by one measurement in all three dimensions. Doubling a sphere's radius multiplies its volume by eight, which is why large bubbles hold so much more air than small ones.
Check radius against diameter
Every one of these formulas uses the radius. A problem giving the diameter needs it halved first, and forgetting to halve produces an answer four or eight times too large. Write the radius down before substituting anything.
Step 2: Try It Yourself
Tap and try it out.
- Diameter6 cm
- Circumference18.85 cm
- Area28.27 cm²
- Circumference ÷ diameter3.14
Circumference is a length, measured in cm. Area is a coverage, measured in cm². The units tell them apart.
Step 3: Watch an Example
One step at a time.
Watch Leo Compare a Cone and a Cylinder
Both have radius 3 and height 10.
- Step 1
The base area is π × 9, about 28.3.
Step 4: Your Turn
Practice makes it stick.
The Base
Problem 1 of 2
A cylinder has radius 5. What is r²?
The Cone
Problem 2 of 2
A cylinder holds 300. A cone with the same base and height holds how much?
Which Formula
1 of 4
A cylinder holds 90. Its matching cone holds?
2 of 4
Radius 4. What is r³?
3 of 4
Diameter 10. What radius goes into the formula?
4 of 4
How many cones of the same base and height fill one cylinder?
Step 5: Quick Check
Show what you know.
Question 1 of 1
A cylinder holds 210. Its matching cone holds?
What You Learned
- A cylinder is its base area times its height.
- A cone with the same base and height is one third of that.
- A sphere is 4/3 πr³, and every formula uses the radius.