page_title: Niccolò Fontana Tartaglia
section_title: Volume of a tetrahedron
hierarchical_section_title: Niccolò Fontana Tartaglia / Volume of a tetrahedron
caption_attribution_description: English: Pyramid from Tartaglia's General Trattato
context_page_description: Niccolò Fontana Tartaglia was an Italian mathematician, engineer, a surveyor and a bookkeeper from the then-Republic of Venice. He published many books, including the first Italian translations of Archimedes and Euclid, and an acclaimed compilation of mathematics. Tartaglia was the first to apply mathematics to the investigation of the paths of cannonballs, known as ballistics, in his Nova Scientia; his work was later partially validated and partially superseded by Galileo's studies on falling bodies. He also published a treatise on retrieving sunken ships.
context_section_description: Tartaglia was a prodigious calculator and master of solid geometry. In Part IV of the General Trattato he shows by example how to calculate the height of a pyramid on a triangular base, that is, an irregular tetrahedron.
  The base of the pyramid is a  triangle , with edges of length , and  rising up to the apex  from points , , and  respectively. Base triangle  partitions into  and  triangles by dropping the perpendicular from point  to side . He proceeds to erect a triangle in the plane perpendicular to line  through the pyramid's apex, point , calculating all three sides of this triangle and noting that its height is the height of the pyramid. At the last step, he applies what amounts to this formula for the height  of a triangle in terms of its sides  (the height from side  to its opposite vertex):
  a formula deriving from the Law of Cosines (not that he cites any justification in this section of the General Trattato).
  Tartaglia drops a digit early in the calculation, taking  as , but his method is sound. The final (correct) answer is:
  The volume of the pyramid is easily gotten after that (not that Tartaglia gives it):
  Simon Stevin invented decimal fractions later in the sixteenth century, so the last figure would have been foreign to Tartaglia, who always used fractions. All the same, his approach is in some ways a modern one, suggesting by example an algorithm for calculating the height of most or all irregular tetrahedra, but (as usual for him) he gives no explicit formula.
caption: 13-14-15-20-18-16 pyramid from the General Trattato di Numeri et Misure, Part IV, Book 2, p. 35.
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