Surface Area & Volume: Six Shapes, Six Formulas, One Strategy
Cylinder, cone, sphere, prism, pyramid — the formulas and the visual cues. Plus the scaling rule that explains why doubling dimensions multiplies volume by 8.
Six shapes, one strategy
The Texas Geometry CBE never asks you to derive a volume formula — it asks you to apply one. Memorize the six formulas, recognize each shape on sight, and you'll handle every 3D problem in under 30 seconds.
The cheat sheet
Formulas to memorize
Any pointy solid (cone, pyramid) holds exactly one third the volume of the corresponding flat-topped solid (cylinder, prism) with the same base and height. Memorize the flat ones, multiply by ⅓ for the pointy ones.
Surface area = sum of every face
Surface area is just the total area of every face you'd paint. For a prism, add up every rectangle. For a cylinder, the lateral surface is a rolled-up rectangle: SAlateral = 2πr · h.
Cylinder volume scaling
A cylinder has radius 4 and height 10. If the radius is tripled but the height stays the same, what happens to its volume?
Open the question →Lateral surface area of a cone
What is the lateral surface area of a cone with r = 7 and slant height ℓ = 10? (use π ≈ 3.14)
Open the question →The k³ scaling rule
Scale every linear dimension by k → volume scales by k³. Halve all dimensions (k = ½) → new volume is (½)³ = ⅛ of original. Triple them (k = 3) → new volume is 3³ = 27× original.
Scale a rectangular prism
A rectangular prism: 6×8×10. If all dimensions are halved, what is the new volume?
Open the question →Composite figure (rectangle + semicircle)
A swimming pool has the shape of a rectangle with a semicircle on each end. Find the perimeter and area.
Open the question →3-second recap
- Prism / cylinder → B · h
- Pyramid / cone → ⅓ B · h (one-third rule)
- Sphere → &frac43;πr³, surface 4πr²
- Surface area = sum of every face's area
- Linear factor k → area × k², volume × k³