A prism or cylinder has volume equal to its base area times its height.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Cones and pyramids are one third
A cone has exactly one third the volume of the cylinder that contains it.
The sphere
V = 4/3 πr³, and its surface area is 4πr².
Scaling a solid
Scaling every length by k multiplies area by k² and volume by k³.
Prisms and pyramids
A prism has volume base area × height. A pyramid on the same base and height has exactly one third of that. The factor of three holds for every base shape, which is a stronger claim than it first appears.
Cylinders, cones and spheres
A cylinder is πr²h, a cone one third of that, and a sphere (4/3)πr³. Cylinder and cone follow the prism-pyramid pattern; the sphere is the odd one out and needs calculus to derive properly.
Perpendicular height, not slant
Every volume formula uses the perpendicular height from base to apex, never the slant height along a face. Slant height belongs in surface area calculations. Substituting one for the other is the standard error.
Radius, not diameter
All the round-solid formulas take the radius. A problem stating the diameter needs it halved before substitution, and forgetting produces an answer four or eight times too large depending on the power.
Step 2: Try It Yourself
Tap and try it out.
- Cubes in one layer4 × 3 = 12
- Layers3
- 4 × 3 × 336 cubic units
Counting every cube gives 36, and so does 4 × 3 × 3. The formula is a shortcut for the count.
Step 3: Watch an Example
One step at a time.
Watch Sam Compare a Cone and a Cylinder
A cylinder and a cone share a radius of 3 and a height of 10.
- Step 1
The cylinder is base area times height: π × 9 × 10, about 282.7.
Step 4: Your Turn
Practice makes it stick.
The Cylinder
Problem 1 of 3
Base area 20, height 7. What is the volume?
cubic units
The Cone
Problem 2 of 3
A cylinder has volume 96. A cone with the same base and height?
cubic units
The Scaling
Problem 3 of 3
A solid is scaled by 3. How many times bigger is its volume?
times
Volumes
1 of 4
Base area 25, height 6. Volume?
2 of 4
A cylinder has volume 150. A cone with the same base and height?
3 of 4
A solid scaled by 2. How many times bigger is the volume?
4 of 4
A solid scaled by 4. How many times bigger is the surface area?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Base area 18, height 9. Volume?
Question 2 of 2
A cone is what fraction of the cylinder containing it?
What You Learned
- Prisms and cylinders are base area times height.
- Cones and pyramids are one third of that.
- Scaling by k multiplies area by k² and volume by k³.