Geometry and Trigonometry SAT Cheat Sheet
The SAT math section frequently tests your ability to recall and apply core geometry and trigonometry concepts quickly. Having a concise, well‑organized reference can save precious seconds on test day and boost confidence when tackling multi‑step problems. Below is a comprehensive cheat sheet that covers the most essential formulas, definitions, and strategies you need to master geometry and trigonometry for the SAT And it works..
It sounds simple, but the gap is usually here.
1. Core Geometry Foundations
1.1. Triangles
- Pythagorean Theorem: a² + b² = c² (right triangles only)
- Special Right Triangles:
- 45°‑45°‑90°: legs are equal; hypotenuse = leg × √2
- 30°‑60°‑90°: short leg = x; long leg = x√3; hypotenuse = 2x
- Triangle Area: ½ × base × height (use any side as base)
- Heron’s Formula (for any triangle with sides a, b, c):
[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)},\quad s = \frac{a+b+c}{2} ] - Triangle Inequality: The sum of any two sides must be greater than the third side.
1.2. Quadrilaterals
- Parallelogram: Opposite sides parallel and equal; opposite angles equal; diagonals bisect each other.
- Rectangle: All angles 90°; diagonals equal. Area = length × width.
- Square: All sides equal; all angles 90°; diagonals intersect at 90° and bisect each other. Area = side².
- Rhombus: All sides equal; opposite angles equal; diagonals perpendicular bisectors. Area = (d₁ × d₂)/2.
- Trapezoid: One pair of parallel sides (bases). Area = ½ × (base₁ + base₂) × height.
1.3. Circles
- Circumference: C = 2πr or C = πd
- Area: A = πr²
- Arc Length: L = (θ/360) × 2πr, where θ is the central angle in degrees
- Sector Area: A = (θ/360) × πr²
- Chord Properties: Equal chords subtend equal angles at the center; the perpendicular from the center bisects the chord.
- Tangent‑Radius Theorem: A radius drawn to a point of tangency is perpendicular to the tangent line.
1.4. Polygons
- Sum of Interior Angles: (n‑2) × 180°, where n = number of sides
- Each Interior Angle of a Regular Polygon: [(n‑2) × 180°] / n
- Exterior Angle Sum: Always 360° (one exterior angle per vertex)
- Each Exterior Angle of a Regular Polygon: 360° / n
1.5. Coordinate Geometry Basics
- Distance Formula: d = √[(x₂‑x₁)² + (y₂‑y₁)²]
- Midpoint Formula: M = ((x₁+x₂)/2, (y₁+y₂)/2)
- Slope: m = (y₂‑y₁)/(x₂‑x₁)
- Slope‑Intercept Form: y = mx + b (m = slope, b = y‑intercept)
- Point‑Slope Form: y‑y₁ = m(x‑x₁)
- Parallel Lines: Same slope
- Perpendicular Lines: Slopes are negative reciprocals (m₁·m₂ = –1)
2. Trigonometry Essentials for the SAT
2.1. Right‑Triangle Trigonometry (SOH‑CAH‑TOA)
- Sine: sin θ = opposite / hypotenuse
- Cosine: cos θ = adjacent / hypotenuse
- Tangent: tan θ = opposite / adjacent
Remember: The hypotenuse is always the side opposite the right angle.
2.2. Reciprocal Functions
- Cosecant: csc θ = 1 / sin θ
- Secant: sec θ = 1 / cos θ
- Cotangent: cot θ = 1 / tan θ
2.3. Pythagorean Identities
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = csc²θ
2.4. Special Angles (Exact Values)
| Angle (°) | sin | cos | tan |
|---|---|---|---|
| 0 | 0 | 1 | 0 |
| 30 | ½ | √3/2 | 1/√3 |
| 45 | √2/2 | √2/2 | 1 |
| 60 | √3/2 | ½ | √3 |
| 90 | 1 | 0 | undefined |
2.5. Unit Circle Basics
- Coordinates on the unit circle are (cos θ, sin θ).
- Reference angles help determine sign based on quadrant:
- Quadrant I: all positive
- Quadrant II: sin positive, cos/tan negative
- Quadrant III: tan positive, sin/cos negative
- Quadrant IV: cos positive, sin/tan negative
2.6. Law of Sines and Law of Cosines (Used for non‑right triangles)
- Law of Sines: a/sin A = b/sin B = c/sin C
- Law of Cosines: c² = a² + b² – 2ab·cos C (and cyclic permutations)
These formulas appear rarely on the SAT but are useful when a problem gives two sides and an included angle or three sides.
