Volume of 3D Shapes — SAT Math Explained
The amount of three-dimensional space enclosed within a solid figure, measured in cubic units.
The Core Idea
Most volume formulas build on the prism formula (base area × height). Pyramids and cones are 1/3 of the corresponding prism/cylinder. Spheres have their own unique formula derived using calculus.
Key Formulas
V = length × width × height
V = side³
V = πr²h
V = (1/3)πr²h
V = (1/3) × base area × height
V = (4/3)πr³
Why Cones Are Pyramids
A cone is essentially a pyramid with a circular base. Both equal 1/3 × (base area) × height. If you fill a cone with a matching cylinder, it takes exactly 3 cones to fill the cylinder.
Composite Solids
For composite 3D shapes, calculate volumes of individual components and add (or subtract if a portion is removed)
Surface Area Note
Surface area (the total area of all faces) is different from volume. SA of a rectangular prism = 2(lw + lh + wh). Always check which measurement is asked for.
Common Errors to Avoid
Forgetting the 1/3 in pyramid and cone formulas
Using diameter instead of radius in cylinder and sphere formulas
Confusing surface area with volume
Practice: Volume of 3D Shapes
5 SAT-style questions. Select your answer and get an instant explanation.
Rectangular prism 3×4×5. Volume?
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