Volume & Surface Area

Volume is how much fits inside, measured in cubic units. Surface area is how much wrapping paper you would need, measured in square units.

For anything with the same cross-section all the way up — a box, a cylinder, any prism — volume is just the base area times the height.

• Box: V = lwh
• Cylinder: V = πr²h

Anything that comes to a point holds exactly one third as much:

• Cone: V = ⅓πr²h
• Pyramid: V = ⅓(base area)(height)
• Sphere: V = 4/3 πr³

Surface area means adding up the faces, so a closed cylinder is two circles plus the rectangle that wraps around: 2πr² + 2πrh.

For any solid with congruent parallel cross-sections, volume is V = A_base · h. Cavalieri's principle extends this: two solids of equal height whose cross-sections have equal areas at every level have equal volume, which is why an oblique prism has the same volume as a right prism with the same base and height.

The one-third factor for cones and pyramids is not a coincidence of shape but the integral of a quadratically shrinking cross-section, ∫₀ʰ A(x/h)² dx = Ah/3. Similarly the sphere's 4/3 πr³ and its surface area 4πr² satisfy dV/dr = S, the same derivative relationship as circles.

Surface area must be assembled face by face, and the lateral surface of a cylinder unrolls into a rectangle of width equal to the circumference, 2πr, and height h. For cones the lateral surface uses the slant height ℓ = √(r² + h²) rather than the vertical height, a distinction that mirrors the base-versus-slant issue in triangle area.