Geometry Foundations

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

Angle Relationships

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

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:

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

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

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

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

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)