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Geometry Fundamentals for SAT Math

May 11, 2025

SAT Math Essentials: Geometry Review

Geometry Vocabulary

  • Acute Angle: An angle that measures less than 90 degrees.
  • Acute Triangle: A triangle with all angles less than 90 degrees.
  • Adjacent Angles: Two angles sharing a vertex and a side.
  • Angle: Formed by two rays with a common endpoint.
  • Arc: A curved section of a circle.
  • Area: Number of square units inside a shape.
  • Bisect: Divide into two equal parts.
  • Central Angle: An angle formed by an arc in a circle.
  • Chord: A segment with endpoints on a circle.
  • Circumference: Distance around a circle.
  • Complementary Angles: Two angles whose sum is 90 degrees.
  • Congruent: Identical in shape and size.
  • Coordinate Plane: A grid divided by x and y axes.
  • Diagonal: A line segment between non-adjacent vertices of a polygon.
  • Diameter: A chord passing through the center of a circle.
  • Equiangular Polygon: All angles equal.
  • Equilateral Triangle: A triangle with equal sides and angles.
  • Exterior Angle: Formed outside a triangle or transversal.
  • Hypotenuse: Opposite the right angle in a triangle, longest side.
  • Interior Angle: Inside a triangle or transversal.
  • Isosceles Triangle: Two equal sides.
  • Line: Continues infinitely in both directions.
  • Line Segment: A part of a line between two points.
  • Major Arc: Arc greater than 180 degrees.
  • Minor Arc: Arc less than 180 degrees.
  • Obtuse Angle: An angle greater than 90 degrees.
  • Obtuse Triangle: A triangle with one angle greater than 90 degrees.
  • Ordered Pair: (x, y) on a coordinate plane.
  • Origin: (0,0) on a coordinate plane.
  • Parallel Lines: Lines that never intersect.
  • Parallelogram: A quadrilateral with parallel sides.
  • Perimeter: Distance around a figure.
  • Perpendicular Lines: Intersect to form right angles.
  • Polygon: A closed figure with more than three sides.
  • Pythagorean Theorem: a² + b² = c² for right triangles.
  • Quadrilateral: A polygon with four sides.
  • Radius: Half the diameter of a circle, from center to edge.
  • Rectangle: A parallelogram with right angles.
  • Regular Polygon: All sides and angles equal.
  • Rhombus: A parallelogram with equal sides.
  • Right Angle: Measures exactly 90 degrees.
  • Right Triangle: Has a right angle.
  • Scalene Triangle: No equal sides.
  • Sector: A slice of a circle.
  • Similar Polygons: Equal angles and proportional sides.
  • Slope: Steepness of a line, (y₂-y₁)/(x₂-x₁).
  • Square: A rectangle with equal sides.
  • Supplementary Angles: Sum of 180 degrees.
  • Surface Area: Sum of areas of a solid's faces.
  • Tangent: Touches a circle at exactly one point.
  • Transversal: Intersects two or more lines.
  • Vertex: A point where lines meet.
  • Volume: Number of cubic units inside a solid.

Important Formulas

Area and Volume

  • Circle: A = πr²
  • Rectangle: A = lw
  • Triangle: A = ½bh
  • Cylinder: V = πr²h
  • Rectangle Solid: V = lwh

Angles

  • Naming Angles: By vertex, a number, or three points.
  • Measuring Angles: Notation m∠A.
  • Classifying Angles:
    • Acute: < 90°
    • Right: = 90°
    • Obtuse: > 90° and < 180°
    • Straight: = 180°

Types of Triangles

  • Scalene: No equal sides.
  • Isosceles: Two equal sides.
  • Equilateral: All sides equal.
  • Acute: All angles < 90°.
  • Right: One angle = 90°.
  • Obtuse: One angle > 90°.

Pythagorean Theorem

  • Used for calculating sides in right triangles, a² + b² = c².
  • Pythagorean Triples: Common sets like (3, 4, 5).

Special Triangles

  • 45-45-90 Triangle: Isosceles right triangles, hypotenuse = side√2.
  • 30-60-90 Triangle: Different side ratios, hypotenuse = 2 × short leg.

Circle Geometry

  • Central Angles and Arcs: Arc length = (central angle/360) × 2πr.

Coordinate Geometry

  • Graphing Points: Using (x, y) coordinates.
  • Distance Formula: d = √((x₂-x₁)² + (y₂-y₁)²).
  • Midpoint Formula: Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2).
  • Slope: (y₂-y₁) / (x₂-x₁).

Quadrilaterals and Polygons

  • Sum of Interior Angles: (n-2) × 180° for n-sided polygon.
  • Sum of Exterior Angles: Always 360°.
  • Parallelogram Properties: Opposite sides and angles are equal, diagonals bisect each other.

Solid Figures

  • Perimeter: Sum of all sides.
  • Area: Square units within a figure.
  • Volume: Cubic units within a solid.

Practice Questions

  • The document includes various practice questions and answers to test understanding of these topics.