section_path: Main concepts/Measures: length, area, and volume
section_name: Measures: length, area, and volume
paragraph_text: Measures: length, area, and volume
  
  Length, area, and volume describe the size or extent of an object in one dimension, two dimension, and three dimensions respectively.
  
  In Euclidean geometry and analytic geometry, the length of a line segment can often be calculated by the Pythagorean theorem.
  
  Area and volume can be defined as fundamental quantities separate from length, or they can be described and calculated in terms of lengths in a plane or 3-dimensional space. Mathematicians have found many explicit formulas for area and formulas for volume of various geometric objects. In calculus, area and volume can be defined in terms of integrals, such as the Riemann integral or the Lebesgue integral.
  
  Other geometrical measures include the curvature and compactness.