3D Figures — Volume and Surface Area

ACT Math · Axiom Academy

LESSON 3D Figures: Volume and Surface Area Every formula you need for ACT solid geometry — and the picture behind each one. The most common 3D shape on the ACT — rooms, aquariums, shipping boxes. Volume is how many unit cubes fill it; surface area is the total of its six flat faces. Watch a box fill with cubes, then unfold into a flat net. Cans, pipes, water tanks — the second most common solid on the ACT. Stack up identical circular disks and you get its volume : base area times height h . Unroll its label and the curved side becomes a flat rectangle — that's the lateral surface area . Volume — stacked circular disks Surface area — two lids + the unrolled side The top and bottom lids contribute to the surface. Unroll the label: it's a rectangle of width (the circumference) and height h , so area . Example — radius 3 , height 10 A cone is a cylinder that tapers to a point. The key fact: its volume is exactly one-third of the cylinder with the same base and height. Pour the cone's contents into that cylinder and it fills just a third of the way up. Volume — one-third of the cylinder Total surface area — base + lateral Perfectly round in every direction — a ball. Build it from horizontal cross-sections: circles that are widest at the equator and shrink to nothing at the poles. Summing those disks gives its volume . These formulas are printed on the ACT reference sheet, but memorizing them saves time. Volume — stacked cross-sections Surface area — four great circles

This is the written version of the interactive lesson above. See the full ACT Math course.