<p>This article examines some constructions of conic sections found in a manuscript (UCLA, MS 170/624, fol. 85r) and attributed to Guidobaldo dal Monte. Following centuries of limited access to Greek sources in the Latin West, the sixteenth century saw a revival of interest in conics, spurred by translations of Apollonius and the work of figures such as as Commandino and Maurolico. Like other mathematicians of the period, Guidobaldo developed, or recovered from ancient sources, methods to draw conic sections. In doing so, he addressed both the theoretical needs of mathematics and the practical requirements of gnomonics, astronomy, and mechanics. The folio includes three geometrical constructions of the conic sections and a marginal note: “Del Galileo”, suggesting a possible link with Galileo Galilei. The article analyzes these constructions, highlighting both the theoretical rigor and practical intent of Guidobaldo’s approach. Particular focus is given to the construction of the hyperbola, which plays a key role in the solution of the Apollonian three circles problem, a topic Guidobaldo discussed in correspondence with Galileo. This problem is another example of the scientific relationship between Galileo and Guidobaldo, along with the study of the centers of gravity of solid figures, already described by the same authors in a paper published in 2022: “Galileo Galilei and the centers of gravity of solids: a reconstruction based on a newly discovered version of the conical frustum contained in manuscript UCLA 170/624”.</p>