Math: Data & Problem Solving·Geometry Trig

SAT Math — Geometry and Basic Trigonometry

What to Expect

Geometry accounts for about 15% of SAT Math questions. The test provides a reference sheet with key formulas (area, volume, special right triangles), but you still need to know when and how to use them.

Area and Perimeter Formulas (provided on test, but memorize anyway)

| Shape | Area | Perimeter/Circumference | |---|---|---| | Rectangle | l × w | 2l + 2w | | Triangle | ½ × base × height | Sum of sides | | Circle | πr² | 2πr (circumference) | | Trapezoid | ½(b₁ + b₂) × h | Sum of sides |

Volume formulas provided:

  • Rectangular prism: V = l × w × h
  • Cylinder: V = πr²h
  • Cone: V = ⅓πr²h
  • Sphere: V = ⁴⁄₃πr³
  • Pyramid: V = ⅓ × base area × height

    • Angle Rules

    Triangles: Angles sum to 180°. Straight line: Angles on one side of a line sum to 180°. Vertical angles: Equal (formed by two intersecting lines). Parallel lines cut by a transversal:

  • Alternate interior angles: equal
  • Corresponding angles: equal
  • Co-interior (same-side interior) angles: supplementary (sum to 180°)

    • Special Right Triangles (Memorize These!)

    30-60-90 triangle: Sides in ratio 1 : √3 : 2

  • Short leg (opposite 30°) = x
  • Long leg (opposite 60°) = x√3
  • Hypotenuse = 2x

    • 45-45-90 triangle: Sides in ratio 1 : 1 : √2

  • Legs = x
  • Hypotenuse = x√2

    • Pythagorean Theorem

    a² + b² = c² (c = hypotenuse)

    Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17 (and multiples: 6-8-10, etc.)

    Basic Trigonometry (SOH CAH TOA)

    In a right triangle:

  • sin θ = opposite / hypotenuse
  • cos θ = adjacent / hypotenuse
  • tan θ = opposite / adjacent

    • Memory aid: SOH CAH TOA

    Example: In a right triangle with hypotenuse 10 and an angle of 30°:

  • Side opposite 30° = 10 × sin(30°) = 10 × 0.5 = 5
  • Side adjacent to 30° = 10 × cos(30°) = 10 × (√3/2) ≈ 8.66

    • The SAT tests trig primarily in the form:

  • "What is the value of sin/cos/tan of a given angle?"
  • "What is the length of a side given an angle and another side?"
  • "Which trig ratio equals a given expression?"

    • Real-world example: A ladder is propped against a wall. The ladder is 20 feet long and makes a 60° angle with the ground. How high does it reach? Height = 20 × sin(60°) = 20 × (√3/2) = 10√3 ≈ 17.3 feet.

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    Key Terms

  • Area: Space inside a 2D figure; measured in square units
  • Volume: Space inside a 3D figure; measured in cubic units
  • Perimeter: Total distance around a 2D figure
  • Pythagorean theorem: a² + b² = c² for right triangles
  • Pythagorean triple: Set of whole numbers satisfying a² + b² = c² (3-4-5, 5-12-13)
  • Special right triangles: 30-60-90 (sides 1:√3:2) and 45-45-90 (sides 1:1:√2)
  • SOH CAH TOA: Mnemonic for sin (opp/hyp), cos (adj/hyp), tan (opp/adj)
  • Vertical angles: Equal angles formed by two intersecting lines
  • Supplementary: Two angles that sum to 180°
  • Transversal: A line that crosses two parallel lines, creating various angle pairs

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Quiz Questions:

Q1. A circle has radius 7. What is its area?

A) 14π B) 49π C) 7π D) 14

Answer: B — Area = πr² = π × 7² = 49π.

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Q2. In a 45-45-90 triangle, each leg has length 6. What is the length of the hypotenuse?

A) 6 B) 6√3 C) 12 D) 6√2

Answer: D — In a 45-45-90 triangle, hypotenuse = leg × √2 = 6√2.

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Q3. A right triangle has legs of 9 and 12. What is the hypotenuse?

A) 15 B) 21 C) √153 D) 18

Answer: A — This is a 3-4-5 triple scaled by 3: (3×3)-(3×4)-(3×5) = 9-12-15. Or use the Pythagorean theorem: 9² + 12² = 81 + 144 = 225 = 15².

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Q4. In a right triangle, the angle opposite the hypotenuse of length 13 is angle A. The side opposite angle A... wait — let's try: In a right triangle with a 30° angle and hypotenuse of 20, what is the length of the side opposite the 30° angle?

A) 10 B) 10√3 C) 20√3 D) 20/√3

Answer: A — sin(30°) = opposite/hypotenuse → opposite = 20 × sin(30°) = 20 × 0.5 = 10. Or recall: in a 30-60-90 triangle, the side opposite 30° is half the hypotenuse.

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Q5. Two parallel lines are cut by a transversal. One angle measures 65°. What is the measure of the co-interior (same-side interior) angle on the same transversal?

A) 65° B) 115° C) 25° D) 90°

Answer: B — Co-interior (same-side interior) angles between parallel lines are supplementary — they sum to 180°. 180° − 65° = 115°.