Surface Area, Volume & Solid Figures
Key formulas, vocabulary, and problem-solving rules for three-dimensional figures.
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Questions Covered in This Set
10 cards to master
Surface area vs. volume (units included)
Surface area = total area of the outside skin, in square units (cm²) — the 'wrapping paper.' Volume = space inside, in cubic units (cm³) — the 'water it holds.'
Volume of any prism or cylinder
V = Bh, where B is the base area and h is the height. For a cylinder that becomes V = πr²h.
Surface area of a cylinder
SA = 2πr² + 2πrh (two circular bases plus the unrolled rectangle of width 2πr and height h).
Volume of a pyramid or cone
V = ⅓Bh — exactly one-third of the prism/cylinder with the same base and height. Cone: V = ⅓πr²h.
Sphere volume and surface area
V = (4/3)πr³ and SA = 4πr².
Height vs. slant height of a cone
Height h is perpendicular from apex to base center; slant height ℓ runs along the surface. They relate by ℓ² = h² + r².
Euler's formula for convex polyhedra
V − E + F = 2. A cube: 8 − 12 + 6 = 2.
Scaling rule for solids
Scale every linear dimension by k: surface area scales by k², volume scales by k³. Double an edge → 4× the paint, 8× the concrete.
Cone with r = 6, slant height 10: find volume
h = √(100 − 36) = 8, so V = ⅓π(36)(8) = 96π ≈ 301.6 in³.
Sphere with volume 288π: find surface area
(4/3)πr³ = 288π → r³ = 216 → r = 6, so SA = 4π(36) = 144π ≈ 452.4 cm².
