Geometry
What this domain covers
Geometry accounts for about 20% of Praxis Core Math. Questions test angle relationships, perimeter and area of two-dimensional figures, volume, the Pythagorean theorem, similar figures, coordinate geometry, and transformations.
Core rules
- Angles: a triangle sums to 180°, a quadrilateral to 360°, a straight line to 180°; vertical angles are equal, and parallel lines cut by a transversal create equal corresponding and alternate interior angles.
- Area: rectangle lw; triangle ½bh; parallelogram bh; trapezoid ½(b₁ + b₂)h; circle πr², with circumference 2πr or πd.
- Volume: rectangular prism lwh; cylinder πr²h. Surface area counts every face.
- Pythagorean theorem: a² + b² = c², where c is the hypotenuse. The converse identifies right triangles.
- Similar figures: corresponding sides share a scale factor k, areas scale by k², and volumes by k³.
- Coordinate geometry: distance = √((x₂ − x₁)² + (y₂ − y₁)²); midpoint = average of the two x-values and of the two y-values.
Worked example
A 6-foot person casts a 4-foot shadow. At the same time a flagpole casts an 18-foot shadow. Similar triangles give 6/4 = h/18, so 4h = 108 and h = 27 feet. The ratio of heights equals the ratio of shadows because the sun angle is the same for both.
Traps that cost marks
- Substituting the diameter for r in πr².
- Assuming that doubling the sides of a figure doubles its area, when area actually grows by k².
- Using a slant edge where the perpendicular height is required in area and volume formulas.
- Forgetting the hypotenuse is always the longest side and opposite the right angle.
- Leaving answers without squared or cubic units.