Ivan Mikhailovich was born in 1786. In 1797, he began service as a 3rd class non-commissioned master in the Yekaterinburg marble scrap expedition, and in 1805 he was transferred to a stone-cutting factory. In 1807, he was promoted to collegiate registrar. In 1810, Ivan Trapeznikov began service at the department of mining affairs in St. Petersburg.

In 1812, he joined the St. Petersburg militia and was wounded at the storming of Polotsk. He received the Order of St. Anna, 4th degree, for his distinguished service. He participated in the foreign campaign. In 1813, he was transferred to the Koporye Infantry Regiment with the rank of lieutenant, and served as a brigade adjutant in the second brigade of the 6th Infantry Division. He received a silver medal on a blue ribbon for his participation in the campaign of 1812. In 1817, he attained the rank of staff captain, and that of captain the following year.

In 1820, Trapeznikov was dismissed from the service "after receiving a wound, in the rank of major in uniform and with a full salary pension." In 1822, he was appointed as an official of special orders for the Yaroslavl Treasury Chamber. From 1825, he served in the Railway Engineers Corps.

In 1831, he married Mariya Nikolayevna Bychkova and received an estate in Vladimir province as a dowry. In 1832, their daughter Yekaterina was born, followed by Lidiya in 1833.

In 1836, Trapeznikov was granted a nobility title.

In 1841, he married his second wife, Mariya Nikolayevna Kalacheva. The second marriage produced daughters Lidiya and Yuliya, the future grandmother of Andrei Kolmogorov. Trapeznikov died in 1851.