Geometry and Trigonometry
Core ideas
Master triangle geometry (similarity, congruence, Pythagorean theorem, area formulas), circle properties (inscribed angles, tangents, chords, sector area and arc length), coordinate geometry (distance, midpoint, slope, equations of lines and circles), and solid geometry (surface area and volume of prisms, cylinders, cones, spheres). In trigonometry, know the unit circle, radian-degree conversion, exact values at standard angles, the sine and cosine laws, and basic identities such as sin^2 + cos^2 = 1.
Traps to watch
- Degrees versus radians in calculator or exact-value problems.
- Ambiguous case of the sine law (two possible triangles).
- Using diameter instead of radius in area and volume formulas.
- Assuming a figure is to scale; only stated facts count.
- Forgetting that an inscribed angle is half the central angle subtending the same arc.
Worked example
A circle has radius 6 and a central angle of 60 degrees. Arc length = r * theta with theta in radians: 60 degrees = pi/3, so arc = 6 * pi/3 = 2pi. Sector area = (1/2) r^2 theta = (1/2)(36)(pi/3) = 6pi. Converting to radians first is the reliable route.
Rule of thumb
Convert angles to radians before using arc length, sector area, or calculus-style formulas; keep degrees for triangle angle-sum reasoning.