Area and Volume
Every area and volume formula describes how a shape fills space. Switch between seven shapes below — three flat, four solid — and watch the measurements respond as you stretch each dimension.
Most area and volume questions come down to a handful of formulas worth having cold: rectangle area (l × w), triangle area ((1/2)bh), rectangular prism volume (l × w × h), cylinder volume (πr²h), and cone volume ((1/3)πr²h — exactly a third of the matching cylinder, as you can see above).
Surface area is different from volume — it’s the total area of every face of a 3D shape added together, not the space it encloses. For a rectangular prism, that’s 2(lw + lh + wh): each of the three pairs of matching faces, doubled.
Composite figures — shapes built from two or more simpler pieces, like a rectangle with a semicircle attached — are solved by breaking them into pieces you already know how to handle, finding each area separately, then adding (or subtracting, for a cutout) them together.
Two more worth memorizing: a sphere’s volume is (4/3)πr³ — notice it depends only on the radius, not a separate height. A square pyramid’s volume is (1/3)s²h, the same “a third of the matching prism” pattern that connects a cone to its cylinder.