円周率を含む数式
π 4 = arctan  1 {\displaystyle {\frac {\pi }{4}}=\arctan 1}
π 4 = arctan  1 2 + arctan  1 3 {\displaystyle {\frac {\pi }{4}}=\arctan {\frac {1}{2}}+\arctan {\frac {1}{3}}}
π 4 = 2 arctan  1 2 − arctan  1 7 {\displaystyle {\frac {\pi }{4}}=2\arctan {\frac {1}{2}}-\arctan {\frac {1}{7}}}
π 4 = 2 arctan  1 3 + arctan  1 7 {\displaystyle {\frac {\pi }{4}}=2\arctan {\frac {1}{3}}+\arctan {\frac {1}{7}}}
π 4 = 4 arctan  1 5 − arctan  1 239 {\displaystyle {\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}} (マチンの公式)
∫ − 1 1 d x 1 − x 2 = π {\displaystyle \int _{-1}^{1}{\frac {dx}{\sqrt {1-x^{2}}}}=\pi }
∫ − ∞ ∞ d x 1 + x 2 = π {\displaystyle \int _{-\infty }^{\infty }{\frac {dx}{1+x^{2}}}=\pi } (逆正接関数)
∫ − ∞ ∞ e − x 2 d x = π {\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}}\,dx={\sqrt {\pi }}} (ガウス積分)
∮ d z z = 2 π i {\displaystyle \oint {\frac {dz}{z}}=2\pi i} (コーシーの積分定理)
∫ − ∞ ∞ sin  x x d x = π {\displaystyle \int _{-\infty }^{\infty }{\frac {\sin x}{x}}\,dx=\pi }
∫ 0 1 x 4 ( 1 − x ) 4 1 + x 2 d x = 22 7 − π {\displaystyle \int _{0}^{1}{x^{4}(1-x)^{4} \over 1+x^{2}}\,dx={22 \over 7}-\pi } (円周率が22/7より小さいことの証明)