Surface Area and Volume Formulas: Complete Reference for 2026
The solids unit in geometry is mostly formulas. Memorize them and you reliably score in the 80s and 90s. Skip them and the unit collapses. The good news is that there are only about ten formulas worth learning, and most of them follow a small set of patterns once you see how they are built.
This guide is a complete reference: every prism, every pyramid, every roundish solid, plus the way to handle composite figures and the units mistakes that cost points.
The Two Big Patterns
Almost every volume formula is one of these:
Prism / cylinder pattern: volume = base area × height.
Pyramid / cone pattern: volume = (1/3) × base area × height.
Almost every surface-area formula is the sum of:
1. Base area (one or two bases).
2. Lateral area (sides — usually perimeter or circumference × height, or a slant-height variation).
For surface area, drop the cone’s circular top and the hemisphere’s flat circle, because they are joined and hidden. Surface area = lateral cone + hemisphere shell = π · 3 · ℓ + 2π · 9. Slant ℓ = √(9 + 64) = √73. SA = 3π√73 + 18π.
Units Mistakes That Cost Points
Volume uses cubic units (cm³, in³, ft³). Surface area uses square units (cm², in², ft²). Mixing them or forgetting the exponent on the unit costs full credit on most state tests.
Conversion catch: 1 m = 100 cm, but 1 m³ = 1,000,000 cm³ (because (100)³). Forgetting this is the most common conversion error.
The Slant Height vs. Height Trap
For pyramids and cones, height (h) is the perpendicular distance from base to apex. Slant height (ℓ) is along the lateral face from base edge to apex.
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Volume always uses h.
Lateral surface area always uses ℓ.
If a problem gives you h and asks for surface area, compute ℓ first. If it gives ℓ and asks for volume, compute h first.
Use Pythagoras on the appropriate right triangle inside the solid.
A Quick Cheat Sheet
Solid
Volume
Surface Area
Cube
s³
6s²
Rectangular prism
lwh
2(lw + lh + wh)
Cylinder
π r² h
2π r² + 2π r h
Cone
(1/3) π r² h
π r² + π r ℓ
Sphere
(4/3) π r³
4π r²
Hemisphere
(2/3) π r³
3π r² (closed)
Square pyramid
(1/3) s² h
s² + 2sℓ
Triangular prism
(base area) h
2(base area) + (perimeter) h
Tape this inside your geometry binder.
Common Mistakes
Using diameter instead of radius. Half the diameter first.
Mixing units of volume and surface area. Cubic for volume, square for surface area.
Treating slant height as height for volume. Volume uses perpendicular height only.
Forgetting the (1/3) in pyramids and cones. Cone volume is one-third of the same-base-and-height cylinder.
Double-counting hidden surfaces in composite figures. Joined faces are not counted in surface area.
Frequently Asked Questions
Are these formulas on the SAT?
The SAT provides most of them in the reference box, but you save time by knowing them cold.
What’s the most-tested formula?
Cylinder volume on the SAT and ACT; rectangular prism in middle school; sphere volume in AP Calculus.
Do I need to prove these formulas?
Not in standard geometry. AP Calculus derives the sphere and cone formulas with integration.
Why is the cone formula one-third of the cylinder?
Because of the way solids built from triangular cross-sections behave under integration. The 1/3 appears for any pointed solid.
What if the problem uses π = 3 or π = 22/7?
Use whatever the problem tells you. Default to the exact answer (in terms of π) unless asked for a decimal.
Closing Thought
Solid geometry is a memorization unit, but the two patterns (base × height for prisms / cylinders, one-third of that for pyramids / cones) cut the workload in half. Drill the cheat sheet, label units, and never confuse slant height with height. Composite figures look hard but decompose into the same nine solids every time.
For more practice, browse our Geometry worksheets and our full Math Topics library. When you are ready for a structured workbook, our Geometry collection covers surface area and volume in depth.
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