Math & Trig
Volume and surface area formulas: sphere, cone, and rectangular box
Compute packaging and tank volumes plus total surface areas for spheres, right circular cones, and rectangular prisms using PI and SQRT.
3D packaging and hopper geometry in one reference
Industrial hoppers, spherical pressure vessels, and shipping cartons rely on three standard 3D solid pairs: Sphere volume (4/3)*PI()*r^3 and surface area 4*PI()*r^2; Cone volume (1/3)*PI()*r^2*h and total surface area PI()*r*(r + SQRT(r^2 + h^2)); Rectangular prism volume L*W*H and surface area 2*(L*W + L*H + W*H).
Notice in the cone surface formula that SQRT(r^2 + h^2) computes the slant height s (for r = 6 and h = 8, s = 10).
| Solid | Dimensions | Volume Formula | Surface Area Formula |
|---|---|---|---|
| Sphere | r = 6 | (4/3)*PI()*r^3 = 904.78 | 4*PI()*r^2 = 452.39 |
| Cone | r = 6, h = 8 | (1/3)*PI()*r^2*h = 301.59 | PI()*r*(r + SQRT(r^2+h^2)) = 301.59 |
| Box | 5 x 4 x 3 | L*W*H = 60 | 2*(L*W + L*H + W*H) = 94 |
Cone, sphere, and rectangular box formulas
Keep consistent linear units (all cm, all inches, or all meters) so volumes come out in cubic units and surface areas in square units.
=ROUND((1 / 3) * PI() * A2^2 * B2, 2)Sphere volume and rectangular carton surface area
Use 4 * PI() * A2^2 for spherical coating area and 2 * (C2*D2 + C2*E2 + D2*E2) for cardboard sheet area.
=ROUND((4 / 3) * PI() * A2^3, 2)Inputs and interpretation
These are ideal solid geometry formulas for positive dimensions in one unit. Total cone area includes the circular base; lateral area excludes it. Box surface area assumes six closed faces and excludes cardboard flaps, overlaps, cutting waste and material thickness. These formulas do not determine structural or pressure-vessel safety.
Try the example
Use the original values or change them to test your understanding.