3D Figures — Rectangular Solids and Cylinders

GRE Quantitative · Axiom Academy

LESSON 3D Figures — Rectangular Solids and Cylinders Build up the volume, unroll the surface, and derive the diagonal — three solids, three ways of seeing why the formulas work. 1. Rectangular Solids — Volume and Surface Area A rectangular solid (or box) has three dimensions: length l , width w , and height h . Picture the base — an l×w rectangle — and then stack a copy of it upward , one thin layer at a time, until it reaches height h. Each layer has the same base area, so the total volume is just that base area repeated h times. Sum of the 3 pairs of matching faces When l = w = h = s, the same formulas collapse to V = s^3 , SA = 6s^2 , and (as the next step will show) — a cube is just a box where all three stacking directions happen to match. 2. The Space Diagonal, Derived The space diagonal is the longest line you can draw inside a box — corner to opposite corner, straight through the middle. Rather than memorizing , watch it get built from two real right triangles , one after the other, using the box with l=3 , w=4 , h=12 . 3. Cylinders — Volume and Surface Area A cylinder is two circular bases of radius r joined by a curved side of height h . The volume is just the circular base area swept up through h — same idea as the box, with a circle instead of a rectangle. The surface area's tricky part is the curved side: watch it peel off and flatten into a plain rectangle. Circular base area ( r²) × height Two circles + the unrolled curved side

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