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Packet 10: Bonus 8

A famous unsolved problem in geometry asks whether one of these polygons, whose side lengths and area are rational, can have irrational diagonals. For 10 points each:
[10e] Name these five-sided polygons whose angles sum to 540 degrees.
ANSWER: pentagons
[10m] This mathematician’s formula can find a solution for triangles. This mathematician’s namesake formula is used to find the area of a triangle given the semiperimeter and the three side lengths.
ANSWER: Hero of Alexandria [or Heron of Alexandria; or Heron's formula; or Heron's method]
[10h] Pentagon, whose diagonals are all rational, must have this property. Ptolemy's theorem relates the four sides and two diagonals of a quadrilateral with this property.
ANSWER: cyclic quadrilaterals
<Science - Math, Vikram Narasimhan> | VAULT II Packet 10

HeardPPBE %M %H %
3720.27100%73%30%

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Conversion

TeamOpponentPart 1Part 2Part 3TotalParts
Georgia ATexas B100010E

Summary

TournamentEditionMatchHeardPPBE %M %H %
VAULT II at Georgia Tech2026-01-24✓628.33100%100%83%
VAULT II at Eden Prairie HS2026-03-13✓323.33100%100%33%
VAULT II at McGill2026-03-13✓420.00100%75%25%
VAULT II at UCSB2026-03-13✓516.00100%40%20%
VAULT II Northern UK2026-04-24✓313.33100%33%0%
VAULT II Southern UK2026-04-24✓417.50100%50%25%
VAULT II at Harvard2026-04-30✓422.50100%75%50%
VAULT II at MINT2026-06-19✓110.00100%0%0%
VAULT II Stanford Online2026-07-17✓720.00100%100%0%