Geometry and Measurement
What the Domain Covers
This domain tests angle and triangle relationships, congruence and similarity, circle geometry, perimeter/area/volume, coordinate geometry, transformations, right-triangle trigonometry, and unit conversion by dimensional analysis.
Angles and Triangles
A triangle's angles sum to 180°; an n-gon's interior angles sum to (n − 2)·180°. An exterior angle equals the sum of the two remote interior angles. Congruence requires SSS, SAS, ASA, AAS, or HL for right triangles; SSA and AAA do not guarantee congruence. In an isosceles triangle the base angles are congruent, and the bisector of the vertex angle is also a median and an altitude.
Similarity and Scaling
Corresponding sides of similar figures are proportional. If the scale factor is k, then perimeters scale by k, areas by k², and volumes by k³. This single relationship answers most scaling items.
Circles and Solids
An inscribed angle is half its intercepted arc; a central angle equals its arc; an angle inscribed in a semicircle is a right angle. Arc length = (θ/360)·2πr and sector area = (θ/360)·πr². Volumes: prisms and cylinders V = Bh; pyramids and cones V = (1/3)Bh; spheres V = (4/3)πr³.
Trigonometry
sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. Law of sines: a/sin A = b/sin B = c/sin C. Law of cosines: c² = a² + b² − 2ab·cos C.
Worked Example
A sector has radius 6 cm and central angle 120°. Find its arc length and area. Arc length = (120/360)(2π·6) = (1/3)(12π) = 4π cm. Area = (120/360)(π·6²) = (1/3)(36π) = 12π cm². Consistency check: the sector is one-third of the circle, so both the arc and the area should be one-third of the full circumference (12π) and full area (36π). Both are.
Traps That Cost Marks
- Substituting the diameter where a formula requires the radius.
- Mixing units before converting; chain conversion factors as fractions so unwanted units cancel.
- Confusing surface area with volume - doubling the radius quadruples lateral area but multiplies volume by eight.
- Forgetting the SSA ambiguity: two distinct triangles can satisfy the given data.