Obseg je v geometriji dolžina zaprte krivulje, po navadi dvorazsežne ravninske krivulje. Največkrat se govori o obsegu pri geometrijskih likih, čeprav pridejo v poštev tudi druge krivulje, kroga, srčnica. V takšnih primerih se še posebej obravnava dolžina loka krivulje.
Obseg mnogokotnika je vsota dolžin vseh njegovih stranic.
Obseg trikotnika s stranicami dolžin a, b in c je:
![{\displaystyle o=a+b+c\,\!.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37c979d4a3658c8a33e3633ad99e48ecda4a92fe)
Obseg štirikotnika s stranicami dolžin a, b, c in d je:
![{\displaystyle o=a+b+c+d\,\!.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c134be5bcb2c59e7267507c1c77fcdd84e5a7c86)
Obseg enakokrakega trikotnika z osnovnico dolžine b in krakoma dolžine a ter pravokotnika s stranicama dolžin a in b je:
![{\displaystyle o=2a+b\,\!,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4a344b43cf92eba67529448a01fa7b6ebfb4676)
![{\displaystyle o=2(a+b)\,\!.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/908a00ddfb49d6949abba26ec064b842ba923653)
Obseg pravilnega mnogokotnika z n stranicami dolžine a je:
![{\displaystyle o=na\,\!.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc19a0db6435f13e87716a72774e2da1141204fc)
Obseg enakostraničnega trikotnika in kvadrata s stranicami dolžine a je tako:
![{\displaystyle o=3a\,\!,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92edd5f00693342ac84bd76f707e5b24794a887e)
![{\displaystyle o=4a\,\!.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93dc861f7c97466f00fc23ad9a9d198d750f8d91)
Obseg krožnice je dan z njenim premerom d ali s polmerom r:
![{\displaystyle o=\pi d=2\pi r\,\!,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3d049ed50c9b47a36d592c3f4cf82b82141a23f)
oziroma s ploščino kroga S:
![{\displaystyle o=2{\sqrt {\pi S}}\approx 3,544908{\sqrt {S}}\,\!.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8cd5ba56281562b861b8ce09d98c3fcf0cb70af)
Tu je π matematična konstanta pi.
Približki za obseg elipse z glavnima polosema a in b:
(Kepler, 1609)
![{\displaystyle o\approx \pi (a+b)\,\!,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de6a78c9972e94d48cabba20fed4a6429e419997)
(Euler, 1773)
![{\displaystyle o\approx \pi \left[{\frac {a+b}{2}}+{\sqrt {\frac {a^{2}+b^{2}}{2}}}\right]\,\!,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abb6324843f8035c4e8abd8e9b22e34b9c2fd731)
![{\displaystyle o\approx \pi \left[{\frac {3}{2}}(a+b)-{\sqrt {ab}}\right]\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cca12f290b84d4e2ed6746bfcff1e669394a0abf)
ali:
![{\displaystyle o\approx \pi {\sqrt {2(a^{2}+b^{2})-{\frac {1}{2}}(a-b)^{2}}}\,\!.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0bb833c812981ea0f186494620a4b9329960b8f)
Vsak približek je točnejši od predhodnega.
Dobra približka je leta 1914 dal Ramanudžan:
![{\displaystyle o\approx \pi \left[3(a+b)-{\sqrt {(3a+b)(a+3b)}}\right]=\pi (a+b)\left[3-{\sqrt {4-h}}\right]\,\!,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/835354480bc9c1bd90572f45d6846ae662a2235e)
![{\displaystyle o\approx \pi (a+b)\left[1+{\frac {3\left({\frac {a-b}{a+b}}\right)^{2}}{10+{\sqrt {4-3\left({\frac {a-b}{a+b}}\right)^{2}}}}}\right]=\pi \left(a+b\right)\left[1+{\frac {3h}{10+{\sqrt {4-3h}}}}\right]\,\!.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/263256bcbbb9e6485fc8782bcf3435ff8f291018)
kjer je h parameter:
![{\displaystyle h=\lambda ^{2}\,\!,\qquad \lambda ={\frac {a-b}{a+b}}\,\!.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52d42e9fd45156f9ee7dd150d84ee5317e64e163)
Tudi tukaj je drugi približek točnejši. Malo manj točen približek je med letoma 1904 in 1920 dal Lindner:
![{\displaystyle o\approx \pi (a+b)\left[1+{\frac {h}{8}}\right]^{2}\,\!.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bfa0b11eb69b023618b39502e62b5b02ceff845)
Obseg elipse s parametrom λ je:
![{\displaystyle o=\pi (a+b)\left[1+{\frac {\lambda ^{2}}{4}}+{\frac {\lambda ^{4}}{64}}+{\frac {\lambda ^{6}}{256}}+\cdots \right]=\pi (a+b)\left[1+\sum _{n=1}^{\infty }\left({\frac {(2n-2)!}{n!(n-1)!2^{2n-1}}}\right)^{2}\lambda ^{2n}\right]\,\!,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/479ca192a53c3afe8a85967c5bbf875e499f861f)
oziroma s parametrom h:
![{\displaystyle o=\pi (a+b)\left[1+{\frac {h}{4}}+{\frac {h^{2}}{64}}+{\frac {h^{3}}{256}}+{\frac {25h^{4}}{16384}}+{\frac {49h^{5}}{65536}}+\cdots \right]=\pi (a+b)\sum _{n=0}^{\infty }{{1 \over 2} \choose n}^{2}h^{n}\,\!,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc8daa4652a7ac827658f86c041165fa46f1f0ab)
približek pa (Hudsonova enačba, 1917):
![{\displaystyle o\approx \pi (a+b){\frac {64-3h^{2}}{64-16h}}\,\!.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/029047d513933f512a35e11e187650110b3bebc4)
Hudsonovo enačbo po navadi pišejo s parametrom L:
![{\displaystyle L={\frac {h}{4}}={\frac {(a-b)^{2}}{(2(a+b))^{2}}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6990b769b67661e4cb385d2cbcf74204c994169b)
![{\displaystyle o\approx {\frac {\pi }{4}}(a+b)\left[3(1+L)+{\frac {1}{1-L}}\right]\,\!.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f5e0e5716ef938a88692638c969ba742ed43135)