Geometry Review for SAT/ACT and Final Exams

Common Shapes and Formulas

Circle

• Circumference:
  • Formula: ( C = 2\pi r )
  • Example: Radius = 5 ( \Rightarrow C = 2\pi \times 5 = 10\pi )
  • Decimal Approximation: ( 10\pi \approx 31.416 )
• Area:
  • Formula: ( A = \pi r^2 )
  • Example: Radius = 5 ( \Rightarrow A = \pi \times 5^2 = 25\pi )
  • Decimal Approximation: ( 25\pi \approx 78.54 )
• Diameter:
  • Formula: ( d = 2r )
  • Example: Radius = 5 ( \Rightarrow d = 2 \times 5 = 10 )
• Chord vs Diameter: Diameter passes through the center; chord does not.

Square

• Area:
  • Formula: ( A = s^2 )
  • Example: Side length = 8 ( \Rightarrow A = 8^2 = 64 ) square units
• Perimeter:
  • Formula: ( P = 4s )
  • Example: Side length = 8 ( \Rightarrow P = 4 \times 8 = 32 ) units
• Problem Example:
  • Given: Area = 36 square feet
  • Find: Perimeter
  • Solution: ( s = \sqrt{36} = 6 \Rightarrow P = 4 \times 6 = 24 ) feet

Rectangle

• Area:
  • Formula: ( A = l \times w )
  • Example: Length = 10, Width = 5 ( \Rightarrow A = 10 \times 5 = 50 ) square units
• Perimeter:
  • Formula: ( P = 2l + 2w )
  • Example: Length = 10, Width = 5 ( \Rightarrow P = 2 \times 10 + 2 \times 5 = 30 ) units
• Problem Example:
  • Given: Area = 40, Length = 8
  • Find: Perimeter
  • Solution: ( w = \frac{40}{8} = 5 \Rightarrow P = 2 \times 8 + 2 \times 5 = 26 ) units
• Complex Problem Example:
  • Given: Length is 3 more than twice the width, Area = 44
  • Solution Steps:
    • Length = 3 + 2w
    • Equation: ( (3+2w)w = 44 )
    • Solve for ( w: 2w^2 + 3w - 44 = 0 )
    • Factor the quadratic: ( w = 4, l = 11 )
    • Perimeter: ( P = 2l + 2w = 2 \times 11 + 2 \times 4 = 30 ) units

Triangles

• Right Triangle and Pythagorean Theorem:
  • Formula: ( a^2 + b^2 = c^2 )
  • Example: Legs = 3, 4; Hypotenuse = 5
  • Calculations:
    • ( 3^2 + 4^2 = 5^2 \Rightarrow 9 + 16 = 25 \Rightarrow \sqrt{25} = 5 )
• Special Right Triangles:
  • 3-4-5 Triangle
  • 5-12-13 Triangle
  • 7-24-25 Triangle
  • 8-15-17 Triangle
  • 9-40-41 Triangle
  • 11-60-61 Triangle
• Problem Example:
  • Given: Hypotenuse = 10, One leg = 6
  • Find: Other leg
  • Solution: ( a^2 + 6^2 = 10^2 \Rightarrow a^2 + 36 = 100 \Rightarrow a^2 = 64 \Rightarrow a = 8 )

Additional Examples

• Using Special Triangles for Quick Solutions:
  • Multiples of known special triangles (e.g., 6-8-10 from 3-4-5)
• Rectangle Problem:
  • Given: AB = 12, AC = 13
  • Find: Area of rectangle
  • Use Pythagorean theorem or special triangles to find missing side.
  • Solution: Width = 5 ( \Rightarrow A = 12 \times 5 = 60 )

Additional Resources

• YouTube Channel:
  • ACT and SAT Math Videos for additional practice and examples.
  • Links provided in the video or search on YouTube.