Geometry: Volume and Surface

Volume

• Definition: The amount of three-dimensional space occupied by a substance or object.
• Units: Cubic units (e.g., cubic meters (m³), cubic centimeters (cm³), cubic inches (in³))
• Formulas:
  • Rectangular prism: V = l × w × h
  • Cylinder: V = π × r² × h
  • Sphere: V = (4/3) × π × r³
  • Cone: V = (1/3) × π × r² × h

Surface

• Definition: The outer boundary of a three-dimensional object.
• Types:
  • Closed surface: A surface that completely encloses a volume (e.g., sphere, cube).
  • Open surface: A surface that does not enclose a volume (e.g., plane, cylinder without ends).
• Surface Area Formulas:
  • Rectangular prism: SA = 2 × (lw + lh + wh)
  • Cylinder: SA = 2 × π × r × (r + h)
  • Sphere: SA = 4 × π × r²
  • Cone: SA = π × r × (r + l)

Volume

• Volume is the amount of three-dimensional space occupied by a substance or object.
• Volume is measured in cubic units, such as cubic meters (m³), cubic centimeters (cm³), and cubic inches (in³).
• The formula for the volume of a rectangular prism is V = l × w × h.
• The formula for the volume of a cylinder is V = π × r² × h.
• The formula for the volume of a sphere is V = (4/3) × π × r³.
• The formula for the volume of a cone is V = (1/3) × π × r² × h.

Surface

• Surface is the outer boundary of a three-dimensional object.
• There are two types of surfaces: closed and open.
• A closed surface is a surface that completely encloses a volume, such as a sphere or a cube.
• An open surface is a surface that does not enclose a volume, such as a plane or a cylinder without ends.
• The formula for the surface area of a rectangular prism is SA = 2 × (lw + lh + wh).
• The formula for the surface area of a cylinder is SA = 2 × π × r × (r + h).
• The formula for the surface area of a sphere is SA = 4 × π × r².
• The formula for the surface area of a cone is SA = π × r × (r + l).