Frustum (Truncated Cone) Volume and Surface

The volume of a truncated cone - a hopper, bucket, tapered footing or pier, flat-topped stockpile, or round-to-round transition. V = (pi h/12)(D^2 + D d + d^2), reported in ft^3, gallons, and yd^3, plus the slant height sqrt(h^2 + (R-r)^2) and lateral wall area pi(R+r)L. A 6 ft top, 2 ft bottom, 4 ft tall hopper holds 54.5 ft^3 = 407 gal = 2.02 yd^3 with 56 ft^2 of sloped plate - more than the average-diameter guess (50.3 ft^3), because the frustum formula, not the mean diameter, is right. Set d = 0 for a full cone, d = D for a cylinder. End disks, wall thickness, and an offset cone are separate. A takeoff aid; verify against the drawing.

V = (pi h/12)(D^2 + D d + d^2); L = sqrt(h^2 + (R-r)^2), R=D/2, r=d/2; lateral area = pi(R+r)L. d=0 -> full cone, d=D -> cylinder.

The conical frustum volume V = (pi h/12)(D^2 + D d + d^2) and lateral surface pi(R+r)L - standard solid geometry as in Machinery's Handbook (Industrial Press), by name; public domain.

Pure solid geometry, public; the two diameters and the height are user-supplied measurements.

Estimate. AHJ and licensed professional govern.

Field names used by the API: large_diameter_ft, small_diameter_ft, height_ft, volume_ft3, volume_gal, lateral_area_ft2

• Frustum volume V = (pi h/12)(D^2 + D d + d^2); more than the mean-diameter guesssolid geometry
• Slant / lateral L = sqrt(h^2 + (R-r)^2); lateral area = pi(R+r)L (no ends)solid geometry
• Limits d=0 gives a full cone, d=D a cylinder; offset/pyramidal shapes are separatescope of this tile