Радиус

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Радиус (остар. полупречник) наричаме разстоянието от центъра до периферията на окръжност или сфера.

Радиусът обикновено се обозначава с малката латинска буква r. Използва се във формулата за изчисляване на дължината на окръжност и формулата за изчисляване на лице на кръг. Терминът идва от латинската дума radius, която означава „лъч“, но също така и спица на колело.

Формули за окръжности

Изчисляване на радиус от обиколка

Радиусът на окръжност с обиколка C е

r={\frac {C}{2\pi }}.

Изчисляване на радиус от площ

Радиусът на окръжност с площ A e

r={\sqrt {\frac {A}{\pi }}}.

Изчисляване на радиус по три точки

За да изчислим радиуса на окръжност, преминаваща през три точки P₁, P₂, P₃, може да се използва следната формула

r={\frac {|P_{1}-P_{3}|}{2\sin \theta }}

където θ е ъгълът $\angle P_{1}P_{2}P_{3}.$

Формули за правилни многоъгълници

Тези формули важат за правилен многоъгълник с n на брой страни.

Изчисляване на радиус от дължината на страната

Радиусът може да бъде изчислен от дължината на страната s като:

r=R_{n}\,s

където

R_{n}={\frac {1}{2\sin {\frac {\pi }{n}}}}\quad \quad {\begin{array}{r|ccr|c}n&R_{n}&&n&R_{n}\\\hline 2&0.5000000&&10&1.6180340-\\3&0.5773503-&&11&1.7747328-\\4&0.7071068-&&12&1.9318517-\\5&0.8506508+&&13&2.0892907+\\6&1.00000000&&14&2.2469796+\\7&1.1523824+&&15&2.4048672-\\8&1.3065630-&&16&2.5629154+\\9&1.4619022+&&17&2.7210956-\end{array}}

Формули за хиперкуб

Изчисляване на радиус от дължината на страната

Радиусът на d-измерен хиперкуб със страна s е

r={\frac {s}{2}}{\sqrt {d}}.

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