Geometry and Trigonometry
Core concepts
This domain covers angle relationships, triangle congruence and similarity, the Pythagorean theorem, circle theorems, area and volume formulas, coordinate geometry basics, and right-triangle and unit-circle trigonometry. You must know the sine, cosine, and tangent ratios, the exact values at 30, 45, and 60 degrees, the Pythagorean identity, the law of sines, and the law of cosines. Radian measure and arc length (s = rθ) and sector area (A = ½r²θ) appear frequently.
Traps to watch
- The law of sines has an ambiguous case (SSA) that can yield zero, one, or two triangles.
- Similar figures scale lengths by k but areas by k² and volumes by k³.
- Inscribed angles are half the intercepted arc; central angles equal the arc.
- Degrees versus radians: mixing them in one calculation is a classic error.
- A triangle with sides 3, 4, 5 is right; 3, 4, 6 is not - verify before applying right-triangle rules.
Rule of thumb
When a problem gives two sides and the included angle, use the law of cosines; when it gives two angles and a side, use the law of sines.
Worked example
A sector has radius 6 and central angle 2π/3 radians. Arc length s = rθ = 6 × 2π/3 = 4π. Area A = ½r²θ = ½ × 36 × 2π/3 = 12π. If the angle were given as 120 degrees, convert first: 120 × π/180 = 2π/3.