🎯 Learning Objectives
- Every area, perimeter, and volume formula the SAT expects you to know.
- How to break composite shapes into simple pieces.
- How surface area differs from volume: and when each appears.
- Speed shortcuts for cylinders, cones, and spheres.
- 10 SAT-style practice questions.
1. Area and Volume Formulas
The SAT provides a reference sheet at the start of every Math section, but you'll save valuable time on test day if you have these committed to memory. Here are the absolute must-knows.
2D Shapes: Area
Shape | Area Formula | Tip |
Rectangle | length × width | Same as base × height |
Triangle | (1/2) × base × height | Height is perpendicular to base |
Parallelogram | base × height | Same as rectangle |
Trapezoid | (1/2)(b₁ + b₂) × h | Average the two bases |
Circle | π × r² | r = radius (half the diameter) |
2D Shapes: Perimeter / Circumference
Shape | Perimeter Formula |
Rectangle | 2(length + width) |
Triangle | Sum of all three sides |
Circle (circumference) | 2π × r = π × d |
3D Shapes: Volume
Shape | Volume Formula | Tip |
Cube | side³ | All edges equal |
Rectangular box | length × width × height | Same as area of base × height |
Cylinder | π × r² × h | Area of circular base × height |
Cone | (1/3) × π × r² × h | Exactly 1/3 the cylinder it fits inside |
Sphere | (4/3) × π × r³ | Famous formula |
Pyramid | (1/3) × base area × h | Same 1/3 strategy as cone |
💡 Pro Tip: The 1/3 Family
Both cones and pyramids are 1/3 of the prism (cylinder or box) they fit inside. Memorize this rule once and you cover both formulas.
2. Composite Shapes: Break and Conquer
If a figure isn't on the formula sheet, it's almost certainly a combination of shapes that ARE. Split it into pieces, find each area or volume, and add or subtract.
STEP 1: Identify the simple shapes inside
STEP 2: Compute each piece separately
STEP 3: Add (combined) or subtract (cut-out)
STEP 4: Double-check units (² for area, ³ for volume)
3. Surface Area vs. Volume
Volume measures how much fits INSIDE a 3D shape (cubic units). Surface area measures how much wrapping paper covers the OUTSIDE (square units).
Common Surface Area Formulas
Shape | Surface Area Formula |
Cube | 6 × side² |
Rectangular box | 2(lw + lh + wh) |
Cylinder | 2πr² + 2πr·h |
Sphere | 4π × r² |
⚠️ Common Mistakes
If a question asks 'how much paint to coat the outside', it wants surface area, not volume.
If it asks 'how much water fits inside', it wants volume.
Wrong unit = automatic wrong answer.
4. Scaling Effects
If you scale a 3D shape's linear dimensions by a factor of k, then:
- Length scales by k
- Surface area scales by k²
- Volume scales by k³
⚡ Strategy: Doubling a Cube
Doubling a cube's side length multiplies its volume by 8 (because 2³ = 8) and its surface area by 4 (because 2² = 4). Tripling multiplies volume by 27 and surface area by 9.
5. Strategies Summary
⚡ Memorize the formulas anyway
Understand what each formula measures and identify its dimensions and units before substitution.
⚡ Use π ≈ 3 for estimating
Quickly checks whether your answer is in the right ballpark.
⚡ For partial figures, use fractions of the whole
A semicircle is 1/2 of a full circle's area, etc.
⚡ For 'water level rises by' problems
Set volume of object = volume of cylinder slice (πr²·Δh).
⚡ For 'painted surface' minus 'window cut-out'
Always subtract the missing piece's area.
6. Summary
📌 Key Takeaways
- 2D area: rect = bh, tri = (1/2)bh, circle = πr².
- Volume: prism = base × h, sphere = (4/3)πr³.
- Cone & pyramid: 1/3 of corresponding prism.
- Composite: split into knowns; add or subtract.
- Scaling: length × k → area × k², volume × k³.
7. Practice: 10 Questions
Q1. A rectangle has length 12 and width 5. Area?
A) 17 B) 30 C) 60 D) 120
Hint: 12 × 5.
Q2. A circle has radius 6. Area?
A) 12π B) 36π C) 6π D) 18π
Hint: πr² = 36π.
Q3. A cylinder has radius 4 and height 10. Volume?
A) 40π B) 80π C) 160π D) 200π
Hint: πr²h = 16 × 10 × π.
Q4. A cube has side 3. Volume?
A) 9 B) 18 C) 27 D) 81
Hint: 3³ = 27.
Q5. A sphere has radius 3. Volume?
A) 4π B) 27π C) 36π D) 108π
Hint: (4/3)π × 27 = 36π.
Q6. A cone has radius 6 and height 5. Volume?
A) 30π B) 60π C) 180π D) 90π
Hint: (1/3)π × 36 × 5 = 60π.
Q7. A triangle has base 8 and height 5. Area?
A) 13 B) 20 C) 40 D) 80
Hint: (1/2)(8)(5) = 20.
Q8. A cube's side is doubled. Its volume becomes…
A) 2× as big B) 4× as big C) 8× as big D) 16× as big
Hint: k³ = 2³ = 8.
Q9. A trapezoid has parallel sides 6 and 10, height 4. Area?
A) 32 B) 24 C) 40 D) 28
Hint: (1/2)(6+10)(4) = 32.
Q10. A circle has circumference 10π. Area?
A) 5π B) 10π C) 25π D) 100π
Hint: C = 2πr → r = 5; A = π × 25.
Answer Key & Worked Solutions
# | Answer | Reasoning |
Q1. | C) 60 | 12 × 5 = 60. |
Q2. | B) 36π | πr² = 36π. |
Q3. | C) 160π | 16 × 10 × π. |
Q4. | C) 27 | 3³. |
Q5. | C) 36π | (4/3)π × 27. |
Q6. | B) 60π | (1/3)π × 36 × 5. |
Q7. | B) 20 | (1/2)(8)(5). |
Q8. | C) 8× as big | k³ rule. |
Q9. | A) 32 | (1/2)(6+10)(4). |
Q10. | C) 25π | r = 5 → πr² = 25π. |