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Geometry Foundations

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1 Introduction to the OAR

About the OAR Who Takes the OAR and Why Study Plan and Test-Day Strategy

2 Math Skills Test (MST)

Arithmetic Fundamentals Fractions, Ratios, and Proportions Percentages and Real-World Applications Algebra Essentials Geometry Foundations Word Problems, Statistics, and Data Interpretation

3 Reading Comprehension Test (RCT)

RCT Overview and Core Strategies Main Idea and Purpose Questions Detail, Fact, and Inference Questions Vocabulary in Context and Author Tone Advanced Reading Strategies and Passage Types RCT Practice Passages and Questions

4 Mechanical Comprehension Test (MCT)

Forces and Newton's Laws of Motion Energy, Work, and Power Simple Machines and Mechanical Advantage Gears, Belts, and Rotational Motion Fluids, Pressure, and Hydraulics Electricity, Circuits, and Magnetism Motion, Heat, and Thermodynamics MCT Practice Problems and Review

5 Full Practice Tests and Final Review

Practice Test: Math Skills Practice Test: Reading Comprehension Practice Test: Mechanical Comprehension Final Review and Test-Day Preparation

Geometry Foundations

Key Takeaways Explain This Topic AI

Geometry questions on the OAR cover angles, triangles, quadrilaterals, circles, and basic 3D shapes.

The sum of interior angles in a triangle is always 180 degrees; in a quadrilateral, always 360 degrees.

Pythagorean theorem (a² + b² = c²) is one of the most frequently tested formulas on the OAR.

Memorize common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25 and their multiples.

Area and perimeter formulas for rectangles, triangles, circles, and trapezoids should be instant recall.

Last updated: March 2026

Geometry questions on the OAR test your ability to apply formulas and spatial reasoning. Some formulas may be provided, but knowing them cold saves time and reduces errors.

  • Angle Fundamentals
  • Types of Angles
  • Type Measure Visual Description
  • Acute Less than 90° Sharp, narrow angle
  • Right Exactly 90° Square corner
  • Obtuse Between 90° and 180° Wide angle
  • Straight Exactly 180° Flat line
  • Reflex Between 180° and 360° More than a straight line
  • Angle Relationships
  • Relationship Rule
  • Complementary Two angles that sum to 90°
  • Supplementary Two angles that sum to 180°
  • Vertical angles Opposite angles formed by intersecting lines — always equal
  • Corresponding angles Same position at parallel line transversals — always equal
  • Alternate interior angles Opposite sides of transversal between parallel lines — always equal
  • Triangles
  • Triangle Angle Sum

The three interior angles of any triangle sum to 180° .

Example: If two angles of a triangle are 45° and 75°, the third angle = 180° - 45° - 75° = 60° .

Triangle Types

  • Type Definition
  • Equilateral All three sides equal, all angles = 60°
  • Isosceles Two sides equal, base angles equal
  • Scalene No sides equal, no angles equal
  • Right One angle = 90°
  • Acute All angles < 90°
  • Obtuse One angle > 90°
  • Triangle Area
  • Area = 1/2 × base × height

The height must be perpendicular to the base. For a right triangle, the two legs serve as base and height.

Example: A triangle has a base of 10 cm and a height of 6 cm.

  • Area = 1/2 × 10 × 6 = 30 cm²
  • The Pythagorean Theorem

For any right triangle with legs a and b and hypotenuse c:

a² + b² = c²

Example: Find the hypotenuse of a right triangle with legs 6 and 8.

  • 6² + 8² = c² → 36 + 64 = c² → c² = 100 → c = 10
  • Common Pythagorean Triples

Memorize these — they appear constantly on the OAR:

  • Triple Multiples
  • 3-4-5 6-8-10, 9-12-15, 12-16-20, 15-20-25
  • 5-12-13 10-24-26, 15-36-39
  • 8-15-17 16-30-34
  • 7-24-25 14-48-50

Speed tip: If you see side lengths that are multiples of a known triple, you do not need to compute. If the legs are 9 and 12, the hypotenuse is 15 (triple: 3-4-5 × 3).

Special Right Triangles

  • Triangle Side Ratios When to Use
  • 45-45-90 1 : 1 : √2 Isosceles right triangles, squares cut diagonally
  • 30-60-90 1 : √3 : 2 Equilateral triangles cut in half
  • 45-45-90 Example: If each leg is 5, the hypotenuse = 5√2 ≈ 7.07

30-60-90 Example: If the shortest side is 4:

  • Side opposite 30° = 4
  • Side opposite 60° = 4√3 ≈ 6.93
  • Hypotenuse (opposite 90°) = 8
  • Quadrilaterals
  • Shape Perimeter Area
  • Rectangle 2(l + w) l × w
  • Square 4s s²
  • Parallelogram 2(a + b) base × height
  • Trapezoid a + b₁ + c + b₂ 1/2 × (b₁ + b₂) × h
  • Rhombus 4s 1/2 × d₁ × d₂ (diagonals)

Example: A rectangular field is 120 meters long and 80 meters wide.

  • Perimeter = 2(120 + 80) = 400 meters
  • Area = 120 × 80 = 9,600 m²
  • Circles
  • Key Circle Formulas
  • Formula Equation
  • Circumference C = 2πr = πd
  • Area A = πr²
  • Diameter d = 2r
  • Arc length (θ/360) × 2πr
  • Sector area (θ/360) × πr²

Use π ≈ 3.14 or π ≈ 22/7 for calculations.

Example: A circular training area has a radius of 50 meters.

  • Circumference = 2π(50) = 100π ≈ 314 meters
  • Area = π(50²) = 2500π ≈ 7,854 m²
  • Volume and Surface Area
  • 3D Shapes
  • Shape Volume Surface Area
  • Rectangular prism l × w × h 2(lw + lh + wh)
  • Cube s³ 6s²
  • Cylinder πr²h 2πr² + 2πrh
  • Sphere (4/3)πr³ 4πr²
  • Cone (1/3)πr²h πr² + πr√(r² + h²)

Example: A cylindrical water tank has radius 3 feet and height 10 feet.

  • Volume = π(3²)(10) = 90π ≈ 282.7 cubic feet
  • Coordinate Geometry Basics
  • Distance Formula
  • Distance = √((x₂ - x₁)² + (y₂ - y₁)²)
  • Example: Distance from (1, 2) to (4, 6) = √((4-1)² + (6-2)²) = √(9 + 16) = √25 = 5
  • Midpoint Formula
  • Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
  • Example: Midpoint of (2, 3) and (8, 7) = ((2+8)/2, (3+7)/2) = (5, 5)
  • Slope
  • Slope = (y₂ - y₁) / (x₂ - x₁)
  • Slope Line Direction
  • Positive Rises left to right
  • Negative Falls left to right
  • Zero Horizontal
  • Undefined Vertical