Geometry — Essential Formulas (Quick Reference)

Every formula and theorem you need for the Texas Geometry CBE — printable for exam day, scannable in under five minutes.

Lines & Angles

• Complementary angles sum to 90°
• Supplementary angles sum to 180°
• Vertical angles are equal
• Parallel lines cut by transversal: corresponding and alternate interior angles are equal; same-side interior angles are supplementary

Triangles

• Interior angles of any triangle sum to 180°
• Exterior angle = sum of the two non-adjacent interior angles
• Triangle inequality: each side < sum of the other two
• Isosceles: two equal sides, two equal base angles
• Equilateral: all 60°, all sides equal

Pythagorean Theorem

• a² + b² = c² (c is hypotenuse)
• Common triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25 (and any multiple)
• 30-60-90: sides in ratio 1 : √3 : 2
• 45-45-90: sides in ratio 1 : 1 : √2

Right-Triangle Trigonometry (SOHCAHTOA)

• sin θ = opposite / hypotenuse
• cos θ = adjacent / hypotenuse
• tan θ = opposite / adjacent = sin θ / cos θ

Area Formulas

• Triangle: ½ × base × height
• Rectangle / parallelogram: base × height
• Trapezoid: ½ × (b₁ + b₂) × height
• Circle: πr²
• Sector: (θ/360) × πr²

Perimeter / Circle

• Circumference: 2πr (or πd)
• Arc length: (θ/360) × 2πr

Volume / Surface Area

• Rectangular prism: V = l × w × h, SA = 2(lw + wh + lh)
• Cylinder: V = πr²h, SA = 2πr² + 2πrh
• Cone: V = ⅓πr²h
• Sphere: V = (4/3)πr³, SA = 4πr²
• Pyramid: V = ⅓ × base area × height

Coordinate Geometry

• Distance: √((x₂ − x₁)² + (y₂ − y₁)²)
• Midpoint: ((x₁ + x₂)/2 , (y₁ + y₂)/2)
• Slope: (y₂ − y₁) / (x₂ − x₁)

Similar Triangles

• Corresponding angles equal, corresponding sides proportional
• If scale factor is k: areas scale by k², volumes by k³

Common Test Mistakes

• Mixing up circumference (2πr) and area (πr²)
• Forgetting Pythagorean only works on right triangles
• Volume of cone is ⅓ of the matching cylinder, not equal
• Reading diameter vs radius on figures

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