Geometry Formulas and Concepts Summary

Definitions and Formulas

• Symbols:
  • P → Perimeter of one base
  • B → Area of base
  • b → base (side)
  • h → Height (altitude)
  • l → Slant Height
  • TA → Total Area / Surface Area (SA)
  • LA → Lateral Area
  • V → Volume

Key Formulas

• Lateral Area:
  $LA = Ph$

• Surface Area:
  $SA = 2B + Ph$

• Volume:
  $V = Bh$

Special Cases: Cylinders and Prisms

• Cylinder:

  • $SA = 2 ext{π}r^2 + 2 ext{π}rh$
  • $V = ext{π}r^2h$

• Right Prism:

  • Volume = $Bh$

Properties and Terms

• Altitude: Segment perpendicular to the parallel planes (height).
• Lateral Faces: Not the bases; joining sides are lateral edges.
• Total Area (TA): Sum of all faces.

Pyramids

• Lateral Area of Regular Pyramid:
  $LA = 1/2 Pl$

• Total Area of Regular Pyramid:
  $TA = B + Pl$

Geometry Concepts

• Euler’s Theorem: $F + V = E + 2$ where

  • F = Faces
  • V = Vertices
  • E = Edges

• Cross Sections: Intersection of a plane and solid yields a polygon.

• Rotating Polygons: Creates three-dimensional figures with circular cross sections.

Example Calculations

• If the circumference of a cylinder is $12 ext{π}$ and the height is 10, find the volume.
• Number of neon tetras allowed in a tank based on volume requirements: each tetra requires 2 gallons
  with 1 gallon = 231 cubic inches.

Important Notes

• Physical Situations: Sometimes volume or surface area is more critical, depending on context (e.g., environmental conditions).
• Use previous examples to apply formulas effectively for problem-solving.