Basic Geometry Review for SAT, HT, and Final Exams

Common Shapes and Formulas

Circle

• Radius (r): Distance from center to any point on the circle.
• Circumference:
  • Formula: (2 \pi r)
  • Example: Radius = 5, Circumference = (2 \pi \times 5 = 10\pi)
  • Decimal Approximation: (10 \times 3.1416 = 31.416)
• Area:
  • Formula: (\pi r^2)
  • Example: Radius = 5, Area = (\pi \times 5^2 = 25\pi)
  • Decimal Approximation: (25 \times 3.1416 = 78.54)
• Diameter:
  • Formula: (2r)
  • Example: Radius = 5, Diameter = (2 \times 5 = 10)

Square

• Area:
  • Formula: (s^2)
  • Example: Side = 8, Area = (8^2 = 64) square units
• Perimeter:
  • Formula: (4s)
  • Example: Side = 8, Perimeter = (4 \times 8 = 32) units
• Finding Perimeter from Area:
  • Given Area = 36, Side = (\sqrt{36} = 6), Perimeter = (4 \times 6 = 24)

Rectangle

• Area:
  • Formula: (l \times w)
  • Example: Length = 10, Width = 5, Area = (10 \times 5 = 50) square units
• Perimeter:
  • Formula: (2l + 2w)
  • Example: Length = 10, Width = 5, Perimeter = (2 \times 10 + 2 \times 5 = 30) units
• Finding Perimeter with Given Area:
  • Given Area = 40, Length = 8:
    • Width (w = \frac{40}{8} = 5)
    • Perimeter = (2 \times 8 + 2 \times 5 = 26)

Solving Problems

Example Problems

1. Length of a rectangle is three more than twice the width. Given area = 44 square cm.

   • Equations: (l = 3 + 2w, l \times w = 44)
   • Solve for (w), then find (l).
   • Perimeter = (2l + 2w)
   • Result: (w = 4, l = 11, P = 30)

2. Length is three more than width, perimeter = 26.

   • Simplify to (l + w = 13)
   • Solve for (w), then find (l).
   • Area = (l \times w)
   • Result: (w = 5, l = 8, A = 40)

Triangles

Right Triangles and Pythagorean Theorem

• Formula: (a^2 + b^2 = c^2)
• Common Triples:
  • (3, 4, 5)
  • (5, 12, 13)
  • (7, 24, 25)
  • (8, 15, 17)
  • (9, 40, 41)
  • (11, 60, 61)
• Examples:
  • Given legs 3 and 4, Hypotenuse = 5
  • Given Hypotenuse 13, one leg 5, find other leg = 12

Example Problems

1. Rectangle ABCD:
   • Given AB = 12, diagonal AC = 13, find area.
   • Use (5, 12, 13) triangle to find missing side = 5.
   • Result: Area = (12 \times 5 = 60)

Additional Resources

• ACT/SAT Prep:
  • Search for ACT and SAT math videos for practice problems and examples.

Conclusion

• Memorizing special right triangles and understanding the basic geometric formulas can significantly save time during exams.