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GACE Middle Grades Mathematics (4-8) (704) study materials
GACE Middle Grades Mathematics (4-8) (704) study resources

GACE Middle Grades Mathematics (4-8) (704): Geometry and Measurement

GACE Middle Grades Mathematics (4-8) (704) · Work through your exam, one topic at a time.

All guide sections

Geometry and Measurement

Geometry and Measurement

Geometry items on GACE 704 combine shape properties with measurement formulas and coordinate work. You should be able to justify angle and triangle relationships, choose the right formula with correct units, and move between a figure, its coordinates, and its image under a transformation.

Core rules to hold

  • Angles: vertical angles are equal; angles on a straight line sum to 180 degrees; angles about a point sum to 360 degrees. When parallel lines are cut by a transversal, alternate interior and corresponding angles are equal, and same-side interior angles are supplementary.
  • Triangles: the interior angles sum to 180 degrees, and an exterior angle equals the sum of the two remote interior angles. Congruence needs SSS, SAS, ASA, or AAS; SSA does not guarantee congruence. Similarity follows from AA, SSS, or SAS similarity, and corresponding sides are proportional.
  • Scale factors: if the linear scale factor is k, areas scale by k^2 and volumes by k^3.
  • Pythagorean relationship: a^2 + b^2 = c^2 for right triangles, with c the hypotenuse; its converse tests for right angles.
  • Circles: circumference = 2(pi)r, area = (pi)r^2. Prisms and cylinders: V = Bh. Cones and pyramids: V = Bh/3. Spheres: V = 4(pi)r^3/3, SA = 4(pi)r^2.
  • Transformations: translations and rotations preserve congruence; dilations preserve shape but change size; reflections reverse orientation.

Worked example

A right circular cylinder has radius 3 cm and height 10 cm. Volume = (pi)r^2h = (pi)(9)(10) = 90(pi), about 283 cubic centimeters. Total surface area is the two circular bases plus the lateral rectangle: 2(pi)r^2 + 2(pi)rh = 18(pi) + 60(pi) = 78(pi), about 245 square centimeters. If the radius were doubled to 6 cm while the height stayed 10 cm, the volume would become (pi)(36)(10) = 360(pi), exactly four times larger, because volume scales with the square of the linear dimension when height is fixed.

Traps that cost marks

  • Substituting the diameter where the formula calls for the radius.
  • Using the slant height of a cone in place of its perpendicular height.
  • Reporting area in linear units or volume in square units.
  • Assuming a figure is drawn to scale or that an angle is right because it looks right.
  • Mixing up the order of corresponding vertices when naming similar or congruent triangles.