ACT Math · High-Frequency Domain (12–14 Questions, ~23% of Math Score)
Plane Geometry & Triangle Relationships
Key ACT Geometry Formula Rule
Unlike the SAT, the ACT does NOT provide a formula reference sheet at the start of the Math section! You must memorize: (1) Area of a triangle (½bh), (2) Pythagorean theorem (a² + b² = c²), (3) Special right triangles (30°-60°-90° and 45°-45°-90°), and (4) Similar figure area ratio (k² = (s&sub1; / s&sub2;)²).
1. The Four Non-Negotiable ACT Geometry Rules
Geometry accounts for nearly a quarter of all ACT Math points. The test questions emphasize these core relationships:
Similar Figures & Area Ratios
If two shapes are similar with linear scale factor k, their perimeter ratio is k, their area ratio is k², and their volume ratio is k³.
Parallel Lines & Transversals
When parallel lines are cut by a transversal, all acute angles are equal, all obtuse angles are equal, and any acute + obtuse angle = 180°.
Special Right Triangles
30°-60°-90° sides are x, x√3, 2x. 45°-45°-90° sides are x, x, x√2. Memorize these to avoid slow trigonometry.
Circle Arcs & Sectors
A circle slice is simply a fraction: (θ / 360°) = (Arc Length / 2πr) = (Sector Area / πr²).
Unless an ACT problem explicitly specifies that a diagram is drawn to scale, never assume an angle is 90° or that two lines are parallel based purely on visual inspection. Look for explicit perpendicular symbols (⊥), angle arc values, or given parallel arrows (||).
2. Rapid Triangle Angle Rules
- Interior Angle Sum: The three interior angles of any triangle always sum to 180°.
- Exterior Angle Theorem: An exterior angle of a triangle equals the sum of the two non-adjacent opposite interior angles (d = a + b).
- Third Side Rule (Triangle Inequality): The third side of a triangle with sides a and b must be strictly between |a - b| and a + b.
Worked Official-Style ACT Example
Question: In a circle with center O, radius r = 12 cm, and central angle ∠AOB = 60°, what is the area, in square centimeters, of sector AOB?
View Step-by-Step Solution
Correct Answer: B
Step 1: Calculate Total Circle Area: Area = πr² = π(12²) = 144π.
Step 2: Find Sector Fraction: 60° / 360° = 1/6.
Step 3: Multiply: Sector Area = 1/6 × 144π = 24π square centimeters.
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