Cylinders extend a circular base through a perpendicular height, while spheres measure every point from one center. Radius must be identified before any area or volume formula is used, especially when the problem supplies diameter or circumference instead.
Right-cylinder anatomy
Right-cylinder anatomyA right cylinder has congruent parallel circular bases of radius 5 units, perpendicular height 12 units between their centers, and a curved lateral surface.Right-cylinder anatomyA right cylinder has congruent parallel circular bases of radius 5 units, perpendicular height 12 units between their centers, and a curved lateral surface.
Figure not drawn to scale.
Right cylinder
A solid with two congruent parallel circular bases whose centers are joined by a perpendicular altitude h. Each base has radius r.
Cylinder quantities
Quantity
Formula
What is included
Units
Lateral area
\(2\pi rh\)
Curved surface only
square units
Total area
\(2\pi rh+2\pi r^2\)
Curved surface and two bases
square units
Volume
\(\pi r^2h\)
Circular base area times height
cubic units
Cylinder lateral area
\[2\pi rh\]
The curved surface unwraps to a rectangle of width 2πr and height h.
Cylinder total surface area
\[2\pi rh+2\pi r^2\]
Add two exposed circular bases to lateral area.
Cylinder volume
\[V=\pi r^2h\]
This is B times h with circular base area B=πr².
Why lateral area is 2πrh
Cylinder and its netA cylinder of radius r and height h unwraps into a rectangle of height h and width equal to circumference 2 pi r, plus two circular bases.Cylinder and its netA cylinder of radius r and height h unwraps into a rectangle of height h and width equal to circumference 2 pi r, plus two circular bases.Worked example
Cylinder area audit
A closed cylinder has radius 5 cm and height 12 cm. Find its lateral and total surface areas.
Lateral surface
LA=2π(5)(12)=120π cm².
Two bases
Two circular bases contribute 2π(5²)=50π cm².
Lateral area is 120π cm²; total surface area is 170π cm².
From circumference to volume
Circumference-first workflow
Recover radius
Use r=C/(2π).
Square radius
Calculate circular base area B=πr².
Multiply by height
Use V=πr²h and attach cubic units.
Spheres and great circles
Planes intersecting a sphereThree panels show a tangent plane meeting a sphere at one point, an offset plane producing a smaller circular section, and a plane through the center producing a great circle.Planes intersecting a sphereThree panels show a tangent plane meeting a sphere at one point, an offset plane producing a smaller circular section, and a plane through the center producing a great circle.
Sphere
The set of all points in space at one fixed distance r from a center.
Great circle
A circular cross-section whose plane passes through the sphere's center. Its radius equals the sphere's radius.
Cylinder and sphere formula comparison
Solid
Surface area
Volume
Critical input
Cylinder
\(2\pi rh+2\pi r^2\)
\(\pi r^2h\)
r and perpendicular h
Sphere
\(4\pi r^2\)
\(\tfrac43\pi r^3\)
radius r
Sphere surface area
\[SA=4\pi r^2\]
Sphere volume
\[V=\frac{4}{3}\pi r^3\]
Worked example
Sphere from diameter
A sphere has diameter 12 m. Find its exact volume.
Convert diameter
Radius is 6 m, not 12 m.
Use the volume formula
V=(4/3)π(6³)=288π m³.
The sphere volume is 288π m³.
Hemispheres and joined solids
Cylinder with a hemisphere capA right cylinder of radius 4 units and height 10 units is capped by a hemisphere with the same radius; the circular contact surface is internal.Cylinder with a hemisphere capA right cylinder of radius 4 units and height 10 units is capped by a hemisphere with the same radius; the circular contact surface is internal.
A cylinder of radius 4 ft and height 10 ft is capped by a hemisphere of radius 4 ft. Find total volume.
Cylinder
π(4²)(10)=160π ft³.
Hemisphere
(2/3)π(4³)=128π/3 ft³.
Total volume is 608π/3 ft³.
Sphere inscribed in a cylinder
Sphere inscribed in a cylinderA sphere of radius r touches both circular bases and the lateral surface of its cylinder, so cylinder radius is r and cylinder height is the sphere diameter 2r.Sphere inscribed in a cylinderA sphere of radius r touches both circular bases and the lateral surface of its cylinder, so cylinder radius is r and cylinder height is the sphere diameter 2r.
Exact inscribed relationship
Sphere
Radius r and volume (4/3)πr³
Cylinder
Radius r, height 2r, and volume 2πr³
Ratio
Sphere volume divided by cylinder volume is 2/3.
Hollow and drilled cylinders
Coaxial cylindrical holeA cylinder of outer radius 6 units and height 9 units has a coaxial cylindrical hole of radius 2 units drilled through its entire height.Coaxial cylindrical holeA cylinder of outer radius 6 units and height 9 units has a coaxial cylindrical hole of radius 2 units drilled through its entire height.