Solids of Revolution Formulas

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=πr2hV = \pi r^2 h

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.

AL=PR⋅LeA_L = P_R \cdot L_e

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=2πr(r+h)A = 2\pi r(r + h)

Centroid of a Cone

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

yˉ=h4\bar{y} = \frac{h}{4}

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=πh(R2−r2)V = \pi h (R^2 - r^2)

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=πr2(hmax+hmin2)V = \pi r^2 \left( \frac{h_{max} + h_{min}}{2} \right)

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=23r2hV = \frac{2}{3} r^2 hAL=2rhA_L = 2 r h

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=13πr2hV = \frac{1}{3}\pi r^2 h

Cone vs Cylinder Volume Ratio

A cone has exactly one-third the volume of a circumscribing cylinder with the same base radius and height.

3.0
6.0
Cylinder Volume169.65
Cone Volume56.55
Ratio (Cone / Cylinder)0.333

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.

AL=πrLA_L = \pi r LA=πr(r+L)A = \pi r (r + L)

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=12πr2hV = \frac{1}{2} \pi r^2 h

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=h3(A1+A2+A1A2)V = \frac{h}{3} (A_1 + A_2 + \sqrt{A_1 A_2})

Frustum of a Cone

A frustum is the portion of a solid that lies between the base and a plane parallel to the base.

4.0
8.0
4.0
Top Radius (r)2.00
Total Cone Volume134.04
Frustum Volume117.29

Formula: (h/3) * (A₁ + A₂ + √(A₁*A₂)) ≈ 117.29

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 $L$.

AL=12(P1+P2)⋅LA_L = \frac{1}{2} (P_1 + P_2) \cdot L
Key Takeaways
  • Cylinders: Volume is base area multiplied by height (V=πr2hV = \pi r^2 h). Oblique cylinders use AL=PRLeA_L = P_R L_e for lateral area.
  • Cones: Volume is exactly one-third of the enclosing cylinder (V=13πr2hV = \frac{1}{3} \pi r^2 h). The centroid is located at h4\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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