Robert William Menmuir, son of James Menmuir, was educated privately. In 1895 he qualified as a surveyor and was elected as a member of the Institution of Chartered Surveyors. He became an associate member of the (British) Institution of Civil Engineers in 1899 and a full member of the Institution in 1916. Arriving in the Cape Colony in 1901 he became town engineer of Woodstock (which was incorporated into Cape Town in 1913) and served as engineer to the Wemmers Hoek Water Scheme. He was a member (with R.O. Wynne-Roberts* and R.H. Charters*) of the Joint Water Committee of Cape Town, which investigated ten schemes to augment the water supply of the city and some of its suburbs. Their final report, with Wynne-Roberts as first author, was published in 1904. In that year Menmuir joined the Cape Society of Civil Engineers, which became the South African Society of Civil Engineers in November 1909. By 1913 he was honorary treasurer of the latter society and in 1917 served as its president. He presented several papers before its members, which were published in the Minutes of Proceedings, including 'Woodstock storm water drainage' (1910) and 'Statistics of the Wemmers Valley watershed' (1912).
Menmuir registered a patent for the clarification of water in 1913. In later years he was the civil engineer of the Silver Mine water supply (from 1923) and consulting engineer for the Groendal water scheme (1928-1932). He also drew up plans of water schemes for Malmesbury (1924), Hopefield (1926) and Uitenhage (1927), plans showing details of Botha River Dam (1927), and drawings for road construction at Fish Hoek (1928).
Menmuir was a foundation member of the South African Association for the Advancement of Science in 1902 and was still a member in 1918. He became a member of the South African Philosophical Society in 1903 and was still a member of its successor, the Royal Society of South Africa, in 1917. In 1910 he was a member of the municipal council of Woodstock. On 26 December 1892 he married Katherine E. Kenyon, with whom he had four children.