Basic Geometry Review for SAT, HT, and Geometry Exams

Common Shapes and Formulas

Circle

• Radius: A line segment from the center of the circle to any point on the circle.
• Circumference:
  • Formula: ( C = 2\pi r )
  • Example: Given radius ( r = 5 ), ( C = 10\pi \approx 31.416 ) (using ( \pi \approx 3.1416 ))
• Area:
  • Formula: ( A = \pi r^2 )
  • Example: ( A = 25\pi \approx 78.54 )
• Diameter:
  • Definition: A line segment that passes through the center and touches two points on the circle.
  • Formula: ( D = 2r )
  • Example: ( D = 10 ) (if ( r = 5 ))

Square

• All sides are equal.
• Area:
  • Formula: ( A = s^2 )
  • Example: ( A = 64 ) (if side ( s = 8 ))
• Perimeter:
  • Formula: ( P = 4s )
  • Example: ( P = 32 )

Rectangle

• Area:
  • Formula: ( A = \text{length} \times \text{width} )
  • Example: ( A = 50 ) (length 10, width 5)
• Perimeter:
  • Formula: ( P = 2L + 2W )
  • Example: ( P = 30 )

Solving Geometry Problems

Example Problem: Perimeter of a Square

• Given: Area ( = 36 ) square feet
• Solution: Find side ( s = \sqrt{36} = 6 ), ( P = 24 ) feet

Circle Problem: Given Circumference

• Circumference ( = 16\pi )
• Solution: ( r = 8 ), ( D = 16 ), Area ( = 64\pi )

Rectangle Problem

• Given: Area ( = 40 ) and length ( = 8 )
• Solution: Width ( = 5 ), Perimeter ( = 26 )

Solving for Missing Dimensions

Problem: Rectangle with Known Perimeter

• Perimeter ( = 26 ), ( L = \text{width} + 3 )
• Solution: Width ( = 5 ), Length ( = 8 ), Area ( = 40 )

Triangles

Right Triangle

• Hypotenuse: Side opposite the right angle.
• Pythagorean Theorem: ( a^2 + b^2 = c^2 )

Special Right Triangles

• Common triangles:
  • 3-4-5
  • 5-12-13
  • 7-24-25
  • 8-15-17
  • 9-40-41
  • 11-60-61
• Multiples: E.g., 6-8-10 is a multiple of 3-4-5

Example Problems

• Finding missing sides using special triangles
• Example: For ( 6, x, 10 ): ( x = 8 )

Additional Practice

• Problem setup and solution for different geometric shapes using known equations.

Additional Resources

• For ACT/SAT math preparation, search for specific practice videos on YouTube.