Overview
SAT 3D geometry problems test the ability to apply volume and surface area formulas to common solids: prisms, cylinders, cones, spheres, and pyramids. The SAT reference sheet provides all five volume formulas, so memorizing volumes is less critical than knowing how to rearrange them to find missing dimensions. Surface area formulas are not on the reference sheet and must be memorized. A key advanced concept is dimensional scaling: when all linear dimensions scale by k, volume scales by k³.
Key Points
1. Volume Formulas (Provided on SAT Reference Sheet)
| Solid | Formula | Notes |
|---|---|---|
| Rectangular prism | V = lwh | l = length, w = width, h = height |
| Cylinder | V = πr²h | r = base radius |
| Cone | V = ⅓πr²h | r = base radius |
| Sphere | V = (4/3)πr³ | r = radius |
| Pyramid | V = ⅓Bh | B = base area, h = height |
These are on the reference sheet — you do not need to memorize them, but you must know how to rearrange them algebraically.
Rearrangement example: Given V = πr²h, V = 100π, h = 4:
r² = V / (πh) = 100π / (4π) = 25
r = 5
2. Surface Area Formulas (NOT on Reference Sheet — Must Memorize)
| Solid | Total Surface Area | Notes |
|---|---|---|
| Rectangular prism | SA = 2(lw + lh + wh) | Three pairs of rectangular faces |
| Cylinder | SA = 2πr² + 2πrh | Two circular bases + lateral rectangle |
| Cone | SA = πr² + πrl | Base circle + lateral surface; l = slant height |
| Sphere | SA = 4πr² | No bases |
Lateral surface area (excludes bases):
- Cylinder lateral: 2πrh
- Cone lateral: πrl (where l = slant height = √(r² + h²))
3. Dimensional Scaling
When all linear dimensions of a solid are multiplied by factor k:
Linear measurements × k
Area / Surface area × k²
Volume × k³
Example: A cylinder has radius 3 and height 5 (V₁ = 45π). Scale every dimension by k = 2:
- New radius = 6, new height = 10
- New volume = π(6²)(10) = 360π = 45π × 2³ ✓
Shortcut: If you know k, multiply original volume by k³ directly — no need to recompute with new dimensions.
4. Cross-Sections of 3D Solids
A cross-section is the 2D shape produced by cutting a solid with a plane.
| Solid | Cut Parallel to Base | Cut Perpendicular to Base |
|---|---|---|
| Cylinder | Circle | Rectangle |
| Cone | Circle (smaller) | Triangle (isosceles) |
| Sphere | Circle | Circle |
| Rectangular prism | Rectangle | Rectangle |
| Pyramid | Smaller similar polygon | Triangle |
The SAT may show a 3D solid and ask what shape the cross-section is, or give a cross-section shape and ask which solid it came from.
5. Problem-Solving Strategy
- Read carefully: underline “volume” or “surface area” — these are different things.
- Identify the solid type.
- Write the appropriate formula.
- Determine whether the given measurement is a radius or a diameter (divide by 2 if diameter given).
- Substitute known values and solve algebraically for the unknown.
- If scaling is involved, apply k² or k³ directly.
Pitfalls and Common Mistakes
Pitfall 1: Confusing surface area with volume The formulas look similar but produce different numerical answers and different units (units² vs. units³). Solving for one when the question asks for the other is a full-question loss. Fix: Before writing any formula, underline the specific quantity the question asks for. Pick the formula that matches.
Pitfall 2: Using diameter when radius is needed Many problems give a diameter measurement. All volume and surface area formulas use the radius. Using d instead of r produces an answer that is 2× (or 4× or 8×) off. Fix: When a circle-based solid is described, immediately halve any diameter to find r before plugging into any formula.
Pitfall 3: Confusing lateral and total surface area “Lateral surface area” excludes the bases. “Total surface area” includes them. Using total when lateral is asked (or vice versa) gives a wrong answer even with the right formula. Fix: Check whether “lateral” or “total” appears in the question. For a cylinder, lateral SA = 2πrh (no base circles); total SA adds 2πr².
Pitfall 4: Using k instead of k³ when scaling volume If a problem says dimensions are scaled by a factor of 2 and asks for the new volume, the answer is 8× the original, not 2×. Fix: Memorize the cube rule: volume scales by k³. Write the exponent explicitly: V_new = V_old × k³.
Related Entries
- Properties_Circles
- Similar_Triangles
- Right_Triangles_Pythagorean
- Coordinate_Geometry
- Radians_Arc_Measure
Quick Reference Card
| Solid | Volume (on sheet) | SA (memorize) |
|---|---|---|
| Rect. prism | lwh | 2(lw + lh + wh) |
| Cylinder | πr²h | 2πr² + 2πrh |
| Cone | ⅓πr²h | πr² + πrl |
| Sphere | (4/3)πr³ | 4πr² |
| Pyramid | ⅓Bh | — (varies by base) |
| Scaling rule | |
|---|---|
| Linear | × k |
| Area / SA | × k² |
| Volume | × k³ |
| Cross-section | Cylinder (parallel to base) → circle; perpendicular → rectangle |