Video narration
Mixing up volume, curved surface area, and total surface area? A cardboard box makes the difference concrete: inside capacity is volume; wrapping is area covering the outside. Notice the families: base area times height, radius for round solids, slant height for side coverage; flat sides use lateral area, not curved. For a cuboid, V=lbh, lateral area=2h(l+b), and TSA=2(lb+bh+lh). For a cube, V=a3, lateral area=4a2, and TSA=6a2. Cylinder: V=πr2h, CSA=2πrh, TSA=2πr(h+r). Cone: V=1⁄3πr2h, CSA=πrl, TSA=πr(l+r). Here h means perpendicular height; l means slant height. Sphere: V=4⁄3πr3 and surface area=4πr2-one measure only. Hemisphere: V=2⁄3πr3, CSA=2πr2, TSA=3πr2. Frustum: V=1⁄3πh(R2+r2+Rr), CSA=π(R+r)l, and TSA adds π(R2+r2). Here's the turn: surface area is not universal. A prism uses V=base area×h, lateral area=base perimeter×h, and TSA=lateral area+2 base areas. A regular pyramid uses V=1⁄3 base area×h, lateral area=1⁄2 base perimeter×l, then adds its base. Choose the purpose: volume fills space, side area wraps it, total area adds the bases.