All of SAT Geometry and Trigonometry

Main ideas, concepts, and lessons

• The video’s goal is a complete, “no filler” review of the core geometry and trigonometry rules needed for the SAT.
• A central message: you must learn these relationships because there’s no “geometry button” on calculators.
• It repeatedly emphasizes that understanding relationships (e.g., angle sums, proportionality, similar triangles, scale factors) is more reliable than memorizing isolated tricks.
• Desmos is encouraged for practice, computation, and verification.
• The speaker also calls out common SAT traps, such as:
  • Confusing altitude measurements
  • Forgetting to square terms
  • Mixing up which angles/sides correspond

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Methodologies / instruction-style content

1) Angle rules (how to determine or use angle relationships)

• Angles between two lines are measured by the opening; corresponding angles repeat if the lines are arranged the same way.
• Straight line rule: If a straight line contains two angles, they sum to 180°.
• Vertical/linear/compound arrangement rule: When a straight line intersects other lines, the angles formed along that straight line must satisfy equality and/or supplementary relationships based on the diagram setup.

• Total interior angles of a polygon

  • Formula: [ \text{Interior angle sum} = 180 \times (\text{number of sides}) - 2 ]

  • Examples:

    • Triangle (3 sides): (180 \times 3 - 2) → interior sum 180° (used as a check)
    • Square (4 sides): (180 \times 4 - 2) → interior sum 360°
  • Key takeaway: Once you know which angles are supplementary (180°) or equal, you can solve for labeled angles by chaining those relationships.

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2) Triangle rules (6 rules the video claims SAT students must know)

• Rule 1: Interior angle sum
  • The three interior angles of any triangle sum to 180°.
• Rule 2: Isosceles triangles
  • If a triangle has two equal side lengths, then it has two equal corresponding angles.
• Rule 3: Equilateral triangles
  • If all three sides are equal, then all three angles are equal.
• Rule 4: Exterior angle
  • An exterior angle equals the sum of the two adjacent interior angles.
• Rule 5: Larger angle ↔ longer opposite side
  • If one angle is larger, then the side opposite it is longer.
• Rule 6: Third side inequality (range)
  • For a triangle with two sides of lengths (a) and (b), the third side (c) must satisfy:
    • (c < a + b)
    • (c > |a - b|)

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3) Right triangles (specific tools and formulas)

• Pythagorean Theorem

  • For legs (a), (b) and hypotenuse (c): [ a^2 + b^2 = c^2 ]

  • Warning: it’s the squared lengths, not the raw values.

  • Special right triangles
  • 45°–45°–90°
    • The legs are equal.
    • If a leg is (s), then the hypotenuse is (s\sqrt{2}).
  • 30°–60°–90°
    • Relative to the shortest side (a):
      • Opposite 30°: (a)
      • Opposite 60°: (a\sqrt{3})
      • Opposite 90°: (2a)
  • Key idea: Once the triangle matches one of these angle patterns, side ratios are fixed and fast to use.

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4) Trigonometry for right triangles: SOHCAHTOA

Use the right-triangle definitions:

• Sine (SOH) [ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} ]

• Cosine (CAH) [ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} ]

• Tangent (TOA) [ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} ]

Worked logic described:

• If you know a value like (\sin(\theta)) equals a fraction from side lengths, you can match it to known angle values.

• The video stresses the distinction:

  • It’s not “angle equals the sine value” — it’s the sine of the angle equals that number.

• Complementary angle theorem (mentioned) [ \cos(\theta) = \sin(90^\circ - \theta) ]

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5) Similarity of triangles (how to set up proportions)

• Meaning of similarity
  • Two shapes (usually triangles) have the same angles, and corresponding side lengths are proportional (not necessarily equal).

• Using proportions to solve side lengths

  • Set up ratios using corresponding sides.
  • Example pattern shown:
    • If (\frac{4}{3} = \frac{x}{6}), then (x = 8).
  • Alternate setups (rearranging fraction positions) can also work.

• Three similarity tests/ways to identify similar triangles

  • SAS (Side-Angle-Side): two sides proportional and the included angle equal
  • AA (Angle-Angle): two angles match
  • SSS (Side-Side-Side): corresponding side pairs proportional

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6) Area and volume (core definitions + SAT formulas)

• Perimeter
  • Sum of side lengths around a 2D shape.
• Area
  • Measures space inside a 2D figure.
  • Given formulas include:
    • Rectangle: (\text{area} = \text{length} \times \text{width})
    • Triangle: (\text{area} = \frac{1}{2} \times \text{base} \times \text{height})
• Surface area
  • Total area of all faces of a 3D figure.
  • SAT often expects you to use formulas rather than derive them.
• Volume
  • Amount of space occupied by a 3D shape.
  • The video implies relevant formulas are provided and should be used directly.

Memorization strategy for surface area formulas

• Repeatedly write surface area formulas (e.g., “5 times”) without looking.
• Suggested time budget: ~10–15 minutes.

Scaling when edge lengths change (volume/surface implications)

• If an edge length doubles:
  • Volume does not just double; apply scale factors.
• Scale factors
  • Linear: (k)
  • Area: (k^2)
  • Volume: (k^3)
• Example:
  • If (k=2), volume increases by (2^3=8).
• Note:
  • Area scale factor also applies to surface area.

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7) Circles (arc/circumference and two main geometry facts)

• Circumference

  • Treated as the perimeter of a circle.

  • Formula: [ C = 2\pi r ]

  • Noted as on the SAT formula sheet.

  • 360° conceptual model
  • A full circle represents 360°, used for arc proportion problems.
  • Arc length via proportion

  • If an arc corresponds to a central angle: [ \frac{\text{arc angle}}{360^\circ} = \frac{\text{arc length}}{\text{circumference}} ]

  • Example:

    • 90° out of 360° with circumference 100:
      • (\frac{90}{360} = \frac{x}{100}) → (x = 25)
  • Tangent-radius perpendicularity
  • A tangent line is perpendicular to the radius at the point of tangency (a 90° angle).
  • Central angle vs. inscribed angle
  • A central angle is twice the inscribed angle (rare but possible on SAT).
  • Requires recognizing the correct configuration.

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8) Altitudes in triangles (formulas + warning about confusion)

• Definition
  • An altitude is a segment drawn from a vertex to the opposite side (or its extension) at a right angle.
• SAT trap emphasized
  • Values you might assume are equal are not always the same; SAT may use similar sub-triangles and distinguish different segments.

• Altitude formula using area [ H = \frac{2 \times \text{area}}{\text{base}} ]

• Altitude formula in right triangles (alternate fast method)

  • For a right triangle with an altitude to a leg, the altitude can be computed using the product of relevant segments divided by the base (example form: (\frac{6 \times 8}{\text{base}})).
  • The base may be found using the Pythagorean Theorem.
• Complex configuration with multiple similar triangles
  • In messy diagrams, drawing the altitude creates multiple triangles that are similar.
  • The method repeats cross-ratio setups (exact labels vary by diagram), such as: [ \frac{x}{z} = \frac{z}{y} = \frac{a}{b} ]

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9) Practice/support resources mentioned

• A “file in the description” contains information from the video for reference.
• An “affiliate” problem set link is provided for practice matching the concepts.
• The video ends with encouragement to practice and do well on the SAT.

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Speakers or sources featured (as stated or clearly implied)

• Speaker/creator: John (referred to as “John” throughout)
• Calculator/tool mentioned: Desmos
• Practice platform/source mentioned: On a Code (with an affiliate link)
• Song/artist reference (brief): Mac Miller (mentioned in the circles section)