In this chapter, we introduce a novel approach to integral and differential calculus in which areas under graphs and tangents to function graphs are determined using elementary geometric constructions. This approach is applied to all standard functions encountered in school mathematics, including power functions (with special attention to y=1/x), exponential and logarithmic functions, as well as the trigonometric functions sine and cosine. The inspiration for this development comes from the Flemish Jesuit Gregorius of St. Vincent, who presented new geometric ideas for determining areas under hyperbolas in his Opus geometricum (1647). Remarkably, these ideas are of a sufficiently general nature to be extended to a much broader class of functions. Building on this insight, we develop a unified geometric framework for determining areas without reducing the problem to purely algebraic antidifferentiation via the fundamental theorem of calculus. The same geometric perspective also enables the determination of tangents using identical tools. In this way, we demonstrate how Gregorius’ method can be placed in a wider mathematical context and reveal its largely unexplored potential.