3-D Shape Formulas
| Geometric Figure | Surface Area | Volume |
|---|---|---|
| Cylinder | ||
| Sphere | ||
| Cone | ||
| Square-based Pyramid | ||
| Rectangular Prism | ||
| Triangular Prism |
Deriving Formulas with Calculus
Surface area is the derivative of volume, and by Fundamental Theorem of Calculus#Fundamental Theorem of Calculus, Part 2 (FTC 2), volume is the integral of surface area.
With the volume
Conversely, with the surface area formula, setting our bounds from
You can think of the volume of the sphere as the surface area of many spheres of increasing radius added together. That is, the volume of any shape is really the Riemann Sum of some surface area.
The surface area of a cylinder has multiple components to it. We should notice that the surface areas are just the Partial Derivatives of the separate variables:
With
Conversely, integrating either of these formulas will give us the volume again:
Integrating the
Integrating the