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.
9 minTEKS 10A,10B,11A,11B,11C,11DGeometry
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
Pyramids and cones are one-third the size of the prism/cylinder they fit inside. Sphere is &frac43; πr³.
Formulas to memorize
Prism / cube: V = B · h (B = base area)Cylinder: V = πr2hPyramid: V = ⅓ B · hCone: V = ⅓ πr2hSphere: V = &frac43; πr3Sphere surface area: SA = 4πr2
The “one third” rule
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 total SA: 2πr² + 2πr · h (two circles + the rectangle)Cone lateral SA: πr · ℓ (where ℓ is the slant height)Cone total SA: πr² + πr · ℓ
Cylinder volume scaling
A cylinder has radius 4 and height 10. If the radius is tripled but the height stays the same, by what factor does the volume increase?
📌 Step 1: Recall how dimensional changes affect volume V = πr²h When r is multiplied by k (and h stays the same): New V = π(kr)²h = k²πr²h = k² × original V 📌 Step 2: Apply k = 3 Factor = 3² = 9 📌 Verification: Original: π(4²)(10) = 160π New: π(12²)(10) = 1440π 1440π / 160π = 9 ✓ 💡 Key rule: Changing ONE linear dimension by factor k changes: • Length → k • Area → k² • Volume → k² (if only radius) or k³ (if all dimensions)
Lateral surface area of a cone
Lateral area of a cone with r=7, l=10? (π ≈ 3.14)
📌 LA=πrl=3.14(7)(10)=219.8
The k³ scaling rule
Half → one-eighth
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 halved, new volume?
📌 Original=480. All halved → V × (1/2)³ = 480/8 = 60
Composite figure (rectangle + semicircle)
A swimming pool has the shape shown below — a rectangle with a semicircle on each end. Find the total area of the pool. (Use π ≈ 3.14)
Rectangle area = 20 × 10 = 200 m². Two semicircles = one full circle with r = 5: π × 5² = 3.14 × 25 = 78.5 m². Total = 200 + 78.5 = 278.5 m².
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
Check yourself
Quick check #1
Find the volume of a cube with edge length 4.
Cube volume formula: V = s³. With s = 4: V = 4³ = 64 cubic units.
Quick check #2
What is the surface area of a sphere with radius 3?
Sphere surface area: A = 4πr². A = 4π(3)² = 4π · 9 = 36π.