Basic Geometry Review for SAT, HT, and Final Exams

Common Shapes and Key Formulas

Circle

• Radius (r): Distance from the center to any point on the circle
• Diameter (D): Twice the radius, D = 2r
• Circumference (C): Distance around the circle, C = 2πr
  • Example: r = 5, C = 2π*5 = 10π ≈ 31.416
• Area (A): Space within the circle, A = πr²
  • Example: r = 5, A = π*5² = 25π ≈ 78.54
• Chord: A line touching two points on the circle but not passing through the center

Square

• Side length (s): Length of a side of the square
• Area (A): Space within the square, A = s²
  • Example: s = 8, A = 8² = 64
• Perimeter (P): Distance around the square, P = 4s
  • Example: s = 8, P = 4*8 = 32
• Example Problem: Given area = 36, find perimeter
  • s² = 36 → s = 6, P = 4*6 = 24

Rectangle

• Length (l): Longer side of the rectangle
• Width (w): Shorter side of the rectangle
• Area (A): Space within the rectangle, A = lw
  • Example: l = 10, w = 5, A = 10*5 = 50
• Perimeter (P): Distance around the rectangle, P = 2l + 2w
  • Example: l = 10, w = 5, P = 210 + 25 = 30
• Example Problem: Given area = 40 and l = 8, find perimeter
  • w = 40/8 = 5, P = 28 + 25 = 26
• Complex Problem: Length is three more than twice the width, area = 44, find perimeter
  • Equation: l = 3 + 2w, A = lw = 44
  • Use substitution and factoring to solve: w = 4, l = 11
  • P = 211 + 24 = 30

Triangles

Right Triangle

• Pythagorean Theorem: a² + b² = c²
• Example 1: Legs = 3 and 4, find hypotenuse
  • a² + b² = 9 + 16 = 25 → c = √25 = 5
• Example 2: Hypotenuse = 13, one leg = 5, find missing leg
  • Known right triangles: 3-4-5, 5-12-13, 7-24-25, etc.
  • a = 5, c = 13, solve for b: 5² + b² = 13² → b² = 169 - 25 = 144 → b = 12
• Special Right Triangles:
  • 3-4-5
  • 5-12-13
  • 7-24-25
  • 8-15-17
  • 9-40-41
  • 11-60-61
• Example 3: Hypotenuse = 10, one leg = 6, find missing leg
  • Scale of 3-4-5: x = 32, y = 42 → 6-8-10 triangle confirms missing side is 8

Application Problems

Finding Missing Sides in Right Triangles

• Example 1: 7-24-25 triangle, missing side is 25
• Example 2: 8-15-17 triangle, missing side is 8
• Example 3: Scaled 3-4-5 triangle: (33, 43 = 9, 12), missing side is 12
• Example 4: Incorrect data
• Example 5: Scaled 5-12-13 triangle (52 = 10, 122 = 24, 13*2 = 26), missing side is 24
• Example 6: Scaled 3-4-5 triangle (310 = 30, 410 = 40), missing side is 50

Rectangle Problem

• Example: Rectangle ABCD with AB = 12 and AC = 13
  • Right triangle: one side 12, hypotenuse 13, missing side 5
  • Area = lw = 125 = 60

Additional Resources

• ACT/SAT Math Videos: Look for more examples and practice problems on YouTube.