Basic Geometry Review for SAT, HT Exams, or Geometry Finals

Common Shapes and Formulas

Circle

• Circumference Formula: ( C = 2\pi r )
  • Example: If radius ( r = 5 ), then ( C = 10\pi \approx 31.416 )
• Area Formula: ( A = \pi r^2 )
  • Example: If radius ( r = 5 ), then ( A = 25\pi \approx 78.54 )
• Diameter: ( D = 2r )
  • Example: If radius ( r = 5 ), then ( D = 10 )
• Chord vs. Diameter: Diameter passes through the center; a chord does not necessarily.

Square

• Area Formula: ( A = s^2 )
  • Example: If side ( s = 8 ), then ( A = 64 ) square units
• Perimeter Formula: ( P = 4s )
  • Example: If side ( s = 8 ), then ( P = 32 ) units
• Example Problem: If ( A = 36 \text{ square feet} ), find ( P )
  • Solution: ( s = 6 ), ( P = 24 ) feet

Rectangle

• Area Formula: ( A = l \times w )
• Perimeter Formula: ( P = 2l + 2w )
• Example: Length ( l = 10 ), Width ( w = 5 )
  • ( A = 50 ) square units, ( P = 30 ) units
• Problem Solving:
  • Given ( A = 40 ) and ( l = 8 ), find ( P )
  • ( w = 5 ), ( P = 26 ) units

Solving Quadratics in Geometry

• Example: Rectangle length ( l = 3 + 2w ), Area = 44
  • Quadratic formed: ( 2w^2 + 3w - 44 = 0 )
  • Factoring: ( (2w + 11)(w - 4) = 0 )
  • Solution: ( w = 4, l = 11 ), ( P = 30 )

Triangles and Pythagorean Theorem

Right Triangles

• Formula: ( a^2 + b^2 = c^2 )
  • ( c ) is the hypotenuse
• Special Triangles:
  • 3-4-5, 5-12-13, 7-24-25, 8-15-17, 9-40-41, 11-60-61
  • Example: ( a = 3, b = 4 ) then ( c = 5 )
• Using Special Triangles for Quick Solutions
  • Recognize multiples of known triangles to solve problems efficiently

Example Problems with Triangles

• 3-4-5 Triangle:
  • Given ( c = 5 ), ( a = 3 ), find ( b )
  • Solution: ( b = 4 )
• 5-12-13 Triangle:
  • Given ( c = 13 ), ( a = 5 ), find missing side
  • Solution: ( b = 12 )
• Multiples of Triangles:
  • If ( a = 6, c = 10 ) (multiples of 3-4-5), then ( b = 8 )

Practice Problems

• Rectangle Problem: Given ( AB = 12 ), ( AC = 13 )
  • Recognize 5-12-13 triangle, find missing side ( BC = 5 )
  • Find area: ( Area = 60 )
• Quick Solutions:
  • Utilizing memorized triangle ratios expedites problem solving on exams like SAT and ACT.