A can’s label unfolds into a rectangle. A conical cap needs part of a disk. Why do the same radius and height give different lateral areas? Compare our own cylinder and cone with r=4 cm and h=3 cm. The key is distinguishing base radius from net radius.
Rotating a figure creates a solid
Prerequisites: arcs and sectors, normals to a plane, sections, prism nets.
We consider right circular cylinders and cones, r>0,h>0. Rotating a filled rectangle about one side creates a cylinder; rotating a right triangle about a leg creates a cone. Rotating only the opposite rectangle side or triangle hypotenuse creates the lateral surface, not the whole volume.
A cylinder’s axis passes through both congruent circular bases’ centers and is perpendicular to their planes. Height h is the distance between them. A generator joins corresponding points of the base circles along the lateral surface; its length equals h.
A cone has one circular base and an apex above its center. The axis passes through apex and center; height is the perpendicular segment joining them. Generator, or slant height, l joins the apex to a point on the base circle. In our cone:
l = √((r)^(2) + (h)^(2)) = √((4)^(2) + (3)^(2)) = 5 cm.
A complete axial section of the cylinder is a 2r×h rectangle; for the cone it is an isosceles triangle with base 2r and equal sides l. Halving the triangle by its altitude gives a right triangle with legs r,h. Do not insert diameter 2r instead of radius r into the slant-height formula.
Cylinder: r=4 cm, h=3 cm
An 8π×3 cm strip and two bases
Cone: axial section
A 288° sector and one base
The label: a rectangle as long as a circumference
Cut the cylinder’s lateral surface along one generator. The rectangular strip has height h and length 2πr, the circle’s circumference, not the disk’s area. Therefore:
S_(cyl,lat) = 2πrh.
Our strip is 8π×3 cm with lateral area 24π≈75.40 cm². A closed cylinder adds two disks of area πr² each: total area 24π+32π=56π≈175.93 cm². The label alone does not need the bases. An overlap strip for gluing requires a separately specified width; the formula does not include it.
The cap: a sector of radius l, not r
A cone cut along a generator unfolds into a sector. Its radius is l=5 cm: the apex-to-rim distance runs along the surface. The sector arc length equals the base circumference, 2πr=8π cm. With central angle θ in radians, lθ=2πr, hence θ=2πr/l=8π/5. In degrees this is 360°·r/l=288°.
A full disk of radius 5 would have rim length 10π, too long for circumference 8π. The disk fraction is r/l=4/5. Its sector area is:
S_(cone,lat) = (r)/(l) π(l)^(2) = πrl.
Thus lateral area is 20π≈62.83 cm². Adding one base gives total area 20π+16π=36π≈113.10 cm². An open cap has no base. Formula πrh would be wrong: vertical height 3 cm is not sector radius 5 cm.
A section is not a net
Unfolding preserves lengths and area along the cut surface. A section retains only points in one plane. Therefore the cone’s triangular axial section cannot replace its sector net. Likewise, a base ellipse in a spatial drawing is the projection of a genuine disk, not a change in its physical shape.
Optional detailThe ordinary cylinder and cone need only the explanation above.
Branch · A cone’s frustum
A cut parallel to our right cone’s base halfway down from the apex removes a similar small cone: radius 2 cm, slant height 2.5 cm. The remaining frustum has R=4,r=2, height 1.5 and slant height s=2.5 cm. Lateral area is the difference 20π−5π=15π cm².
In general, for a right circular frustum S=π(R+r)s. Let the large cone’s full slant height be L and the small one tL, with r=tR. The lateral-area difference is πRL−π(tR)(tL)=πR(1−t²)L=π(R+r)s because s=L(1−t). Total area adds πR²+πr²: here 15π+16π+4π=35π cm². The lateral net is an annular sector of radii 5 and 2.5 cm and angle 288°, not a trapezoid. Its arcs match the two base circumferences. An oblique cut does not automatically meet these conditions.
Sources and limits
Definitions, axial sections and nets: Merzliak and coauthors, 2019, sections 19–20, pp.122–129. Additional derivation checks: AMSI, Cones, Pyramids and Spheres, cylinder and cone surface-area sections. Numbers and diagrams are our own; ideal areas exclude thickness and seams. Volumes follow in G034.