ACT® · Math: Geometry — Geometric Measurement and Modeling
Volume of Cones, Pyramids and Spheres
A cone or pyramid holds exactly one third of the prism or cylinder that wraps around it, and a sphere's volume, four thirds π r cubed, sits exactly between the cylinder and cone that bracket it. No formula sheet is given, so these have to come from memory.
The one third factor in V = (1/3) π r2 h for a cone and V = (1/3) (base area) h for a pyramid comes from the same geometric fact: any pointed shape that tapers to a single apex holds exactly one third of the prism or cylinder with the same base and the same height. Stack three identical cones and you fill the cylinder around them. That factor doesn't care whether the base is a circle, a square, or a triangle — it only cares that the shape narrows to a point.
A sphere sits inside this same family in a tidy way. Take a cylinder with radius r and height 2r, and sit a cone of the same radius and height inside it, apex at the center. The sphere of radius r has a volume exactly two thirds of that cylinder — right between the cone's one third and the cylinder's full volume. That relationship is where V = (4/3) π r3 comes from, and it's worth knowing even without rederiving it each time.
The height that belongs in every one of these formulas is the perpendicular height — straight up from the base to the apex or center, never the slant height along a triangular face. Slant height measures the surface, so it shows up in surface area formulas, not volume. Keeping that distinction straight is most of the battle once the three formulas are memorized.
Key terms
- Vertical height
- The perpendicular distance from the base straight up to the apex, the value volume formulas require.
- Slant height
- The distance from the apex down to the base measured along a lateral face, used in surface area, not volume.
- Apex
- The single point at the top of a cone or pyramid where all lateral surfaces meet.
- Radius vs diameter
- The radius is half the diameter; sphere and cone volume formulas need the radius, so a given diameter must be halved first.
Watch it explained
- How it worksHow does the same one third factor show up in the cone and the pyramid formula for one shared reason?
- How it worksHow does four thirds π r cubed sit between the cylinder and the cone that bracket a sphere?
- Worked exampleFind the volume of a cone with radius 3 and height 7.
- Worked exampleFind the volume of a sphere whose diameter measures 12 cm.
- Common mistakeWhy isn't a pyramid's slant height the height that belongs in its volume formula?
Check yourself
End of topic.