Cylinders and Cones - Theory & Concepts

Solids of Revolution Formulas

Learning Objectives

Calculate the volume and surface area of right and oblique cylinders.
Determine properties of hollow cylinders (pipes) and truncated cylinders.
Compute volume and lateral area for cylindrical ungulas (hoofs).
Calculate the volume, surface area, and centroid location for right circular cones.
Compute volumes and lateral surface areas for frustums of regular pyramids and cones.
Evaluate the volume of a paraboloid of revolution.

These are solid figures generated by revolving a plane area about an axis.

Volume of a Cylinder

The volume of a right circular cylinder is the product of its circular base area and height.

V = \pi r^2 h

Variables

Symbol	Description	Unit
$V$	Volume	-
$r$	Base radius	-
$h$	Height	-

Lateral Area of an Oblique Cylinder

For an oblique (slanted) cylinder, the lateral area requires the perimeter of the right section (perpendicular to the lateral edges) rather than the base.

A_L = P_R \cdot L_e

Variables

Symbol	Description	Unit
$A_L$	Lateral Area	-
$P_R$	Perimeter of the right section	-
$L_e$	Length of the lateral edge	-

Surface Area of a Cylinder

The total surface area of a closed right circular cylinder includes the top and bottom circular bases and the lateral curved surface.

A = 2\pi r(r + h)

Variables

Symbol	Description	Unit
$A$	Total surface area	-
$r$	Base radius	-
$h$	Height	-

Centroid of a Cone

The geometric centroid (or center of volume) of a right circular cone lies on its axis of symmetry.

\bar{y} = \frac{h}{4}

Variables

Symbol	Description	Unit
$\bar{y}$	Distance from the base to the centroid	-
$h$	Perpendicular height	-

Volume of a Hollow Cylinder (Pipe)

For a hollow cylinder or pipe, the volume of the material is the difference between the outer cylinder volume and the inner (hollow) cylinder volume.

V = \pi h (R^2 - r^2)

Variables

Symbol	Description	Unit
$V$	Volume of material	-
$R$	Outer radius	-
$r$	Inner radius	-
$h$	Height or length	-
$t$	Thickness of the wall ( $t = R - r$ )	-

Volume of a Truncated Right Circular Cylinder

A truncated right circular cylinder is formed when a cylinder is cut by a plane inclined to the base. The volume is the product of the base area and the average of the longest and shortest heights.

V = \pi r^2 \left( \frac{h_{max} + h_{min}}{2} \right)

Variables

Symbol	Description	Unit
$V$	Volume	-
$r$	Radius of the circular base	-
$h_{max}$	Maximum height	-
$h_{min}$	Minimum height	-

Volume and Lateral Area of a Cylindrical Ungula (Hoof)

A cylindrical ungula is a wedge-shaped portion of a cylinder cut off by a plane intersecting the base. For a right circular cylinder cut by a plane passing through the diameter of the base, the volume and lateral area are given by specific formulas.

V = \frac{2}{3} r^2 h

A_L = 2 r h

Variables

Symbol	Description	Unit
$V$	Volume	-
$A_L$	Lateral Area	-
$r$	Radius of the base	-
$h$	Height of the ungula (at the highest point)	-

Volume of a Cone

The volume of a right circular cone is one-third of the volume of a cylinder with the same base and height.

V = \frac{1}{3}\pi r^2 h

Variables

Symbol	Description	Unit
$V$	Volume	-
$r$	Base radius	-
$h$	Perpendicular height	-

Interactive Simulation

Explore the relationship between the volume of a cone and the volume of a cylinder using the simulation below.

Surface Area of a Cone

The lateral surface area $A_L$ and total surface area $A$ of a right circular cone, where $L$ is the slant height.

A_L = \pi r L

A = \pi r (r + L)

Variables

Symbol	Description	Unit
$A_L$	Lateral Area	-
$A$	Total Surface Area	-
$r$	Radius	-
$L$	Slant height ( $L = \sqrt{r^2 + h^2}$ )	-

Volume of a Paraboloid of Revolution

A paraboloid of revolution is formed by revolving a parabola about its axis. Its volume is exactly half that of the circumscribing cylinder.

V = \frac{1}{2} \pi r^2 h

Variables

Symbol	Description	Unit
$V$	Volume	-
$r$	Radius of the circular base	-
$h$	Height of the paraboloid	-

Volume of a Frustum of a Pyramid or Cone

A frustum is the portion of a solid that lies between the base and a plane parallel to the base. $A_1$ and $A_2$ are the areas of the two parallel bases.

V = \frac{h}{3} (A_1 + A_2 + \sqrt{A_1 A_2})

Variables

Symbol	Description	Unit
$V$	Volume	-
$A_1$	Lower base area	-
$A_2$	Upper base area	-
$h$	Perpendicular height between bases	-

Interactive Simulation

Use the simulation below to visualize how the properties of a frustum change dynamically.

Lateral Surface Area of a Frustum of a Regular Pyramid or Cone

For a regular frustum, the lateral area is determined using the base perimeters (or circumferences) and the slant height $L$ .

A_L = \frac{1}{2} (P_1 + P_2) \cdot L

Variables

Symbol	Description	Unit
$A_L$	Lateral Area	-
$P_1$	Perimeter/Circumference of lower base	-
$P_2$	Perimeter/Circumference of upper base	-
$L$	Slant height	-

Key Takeaways

Cylinders: Volume is base area multiplied by height ( $V = \pi r^2 h$ ). Oblique cylinders use $A_L = P_R L_e$ for lateral area.
Cones: Volume is exactly one-third of the enclosing cylinder ( $V = \frac{1}{3} \pi r^2 h$ ). The centroid is located at $\frac{h}{4}$ from the base.
Cylindrical Ungula: Formulas for volume and lateral area apply to a cylindrical hoof cut by an intersecting plane.

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