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{"Title": "Volume of the $d$-dimensional unit sphere", "Definition": "$S_d = \\text{vol}(\\{x\\in\\mathbb{R}^{d+1}: |x| = 1\\})$ is the $d$-dimensional volume of the unit sphere in $(d+1)$-dimensional Euclidean space.", "Parameters": {"d": {"display": "$d$", "type": "Z", "constraints": "$d \\geq -1$"}}, "Comments": {}, "Formulas": {"formula-gamma": "$S_d = \\frac{2\\pi^{(d+1)/2}}{\\Gamma((d+1)/2)}$.", "formula-ball": "$S_{d-1} = dB_d$ for $d\\geq 0$, where $B_d$ denotes the $d$-dimensional volume of the unit ball in $\\mathbb{R}^d$."}, "Programs": {}, "References": {}, "Links": {"Wiki": {"title": "Wikipedia: N-sphere", "url": "https://en.wikipedia.org/wiki/N-sphere"}}, "Similar tables": "", "Keywords": "", "Tags": ["elementary geometry", "integral", "period"], "Data properties": {"type": "R", "complete": "no", "rigour": "proven", "rigour details": "Computed in interval or ball arithmetic throughout, so the digits written follow from the width of the result rather than from a guard chosen in advance."}, "Display properties": {"number-header": "$S_d$"}, "Numbers": {"-1": "0", "0": "2", "1": {"number": "6.283185307179586476925286766559005768394338798750211641949889184615632812572417997256069650684234135", "equals": "HREF{Rational_multiples_of_pi#2}"}, "2": "12.56637061435917295385057353311801153678867759750042328389977836923126562514483599451213930136846827", "3": "19.73920880217871723766898199975230227062739881448158125282669875244008964483841048600354680743710446", "4": "26.31894506957162298355864266633640302750319841930877500376893166992011952645121398133806240991613928", "5": "31.00627668029982017547631506710139520222528856588510769414453810380639491746570603756670103260288619", "6": "33.07336179231980818717473607157482154904030780361078154042084064406015457863008644007114776810974527", "7": "32.46969701133414574548011089623503708324252855756180723048928646332361848560906539670728907825099751", "8": 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