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Essential Geometry Formulas for Exam Prep

Apr 24, 2025

Basic Geometry Review for Exams

Circles

  • Circumference Formula: ( C = 2\pi r )
    • Example: Radius ( r = 5 )
    • Circumference: ( 2\pi \times 5 = 10\pi )
    • Decimal approximation: approx. 31.416
  • Area Formula: ( A = \pi r^2 )
    • Example: ( r = 5 )
    • Area: ( \pi \times 5^2 = 25\pi )
    • Decimal approximation: approx. 78.54
  • Diameter Formula: ( D = 2r )
    • Example: ( r = 5 )
    • Diameter: ( 2 \times 5 = 10 )
  • Difference between Diameter and Chord:
    • Diameter passes through the center, connects two points on the circle edge
    • Chord connects two points but does not pass through the center

Squares

  • Area Formula: ( A = s^2 )
    • Example: side length ( s = 8 )
    • Area: ( 8^2 = 64 )
  • Perimeter Formula: ( P = 4s )
    • Example: ( P = 4 \times 8 = 32 )
  • Example Problem: Given area 36, find perimeter
    • Side length ( s = 6 ) (since ( 6^2 = 36 ))
    • Perimeter: ( 4 \times 6 = 24 )

Rectangles

  • Area Formula: ( A = l \times w )
  • Perimeter Formula: ( P = 2l + 2w )
    • Example: ( l = 10, w = 5 )
    • Area: ( 10 \times 5 = 50 )
    • Perimeter: ( 20 + 10 = 30 )
  • Example Problem: Given ( A = 40 ) and ( l = 8 ), find ( w ) and ( P )
    • ( w = \frac{40}{8} = 5 )
    • Perimeter: ( 16 + 10 = 26 )

Special Rectangle Problems

  • Length = 3 more than twice width
    • ( l = 3 + 2w )
    • Example: ( A = 44 ), solve for ( l, w )
    • Factor quadratic equation to find ( w = 4 ), ( l = 11 )
    • Perimeter: ( 30 )

Triangles

  • Right Triangle: Use Pythagorean theorem ( a^2 + b^2 = c^2 )
    • Example: legs ( a = 3, b = 4 ), hypotenuse ( c = 5 )
  • Special Right Triangles:
    • 3-4-5
    • 5-12-13
    • 7-24-25
    • 8-15-17
  • Example: Hypotenuse ( c = 13 ), leg ( a = 5 ), find ( b = 12 )

Problem Solving

  • Rectangle Problem: Given ( l = 3 + w ), ( P = 26 )
    • Solve: ( l = 8, w = 5 )
    • Area: ( 40 )
  • Right Triangle Problems: Find unknown sides using special number sets

Exam Preparation

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

This review provided essential formulas and problem-solving strategies for basic geometric shapes, targeting exam preparation for SAT, ACT, and other tests involving geometry.

Full transcript