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Math · 36 · Geometry and Trigonometry

3D Shapes

Notes

Key points

  • Prism and cylinder: V = base area × height.
  • Cone and pyramid: 13\tfrac{1}{3} of the matching cylinder or prism.
  • Volume formulas are on the reference sheet; surface area formulas are not.
  • Surface area = sum of all face areas — unfold the net.
  • Scale lengths by kk: surface area ×k2\times k^2, volume ×k3\times k^3.
  • Convert units before cubing: 1 ft3=1728 in31\text{ ft}^3=1728\text{ in}^3.

Formulas

Rectangular prismV=ℓwhSA=2(ℓw+ℓh+wh)V=\ell wh\qquad SA=2(\ell w+\ell h+wh)
CylinderV=πr2hSA=2πr2+2πrhV=\pi r^2h\qquad SA=2\pi r^2+2\pi rh
ConeV=13πr2hV=\dfrac{1}{3}\pi r^2h
SphereV=43πr3SA=4πr2V=\dfrac{4}{3}\pi r^3\qquad SA=4\pi r^2
Rectangular pyramidV=13ℓwhV=\dfrac{1}{3}\ell wh
Box diagonald=ℓ2+w2+h2d=\sqrt{\ell^2+w^2+h^2}Not on the reference sheet

Common traps

  • Using the diameter in πr2h\pi r^2h — halve it first.
  • Doubling every length doubles the volume? No: ×23=8\times 2^3=8.
  • Using the slant height in a cone or pyramid volume; it needs the perpendicular height.

SAT tip

Missing dimension from a given volume: copy the reference-sheet formula, plug in, solve.

Test yourself

10 official SAT questions on 3D Shapes, easiest first.

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