This paper suggests one possible approach to demonstrating how to invert the Radon transform, in a language accessible to senior undergraduates in the mathematical sciences, from a new, rigorous point of view that closely parallels actual numerical methods, and involves only calculus. The present approach relies on a simple inversion formula, based on a well-known convolution technique first formulated by Z. H. Cho in 1974, but accomplished by him with numerical approximations only [see \textit{Z. H. Cho} et al., Computerized image reconstruction methods with multiple photon/X-ray transmission scanning, Phys. Med. Biol., 19, 511-522 (1974), p. 517 and \textit{Z. H. Cho}, General views on 3-D image reconstruction and computerized transverse axial tomography, IEEE Trans. Nucl. Sci., NS-21, 44-71 (1974), p. 62]. Briefly, the present method involves an auxiliary function \(G_ c\), which depends on a positive parameter, c; given the Radon transform, \({\mathcal R}f\), of some function, f, the function \(G_ c\) recovers f as follows. \[ Theorem: f(x,y)=\lim_{c\to 0}\frac{1}{\pi}\int^{\pi}_{0}\int^{\infty}_{-\infty}{\mathcal R}f(p-x \cos \alpha -y \sin \alpha,\alpha)G_ c(p)dp d\alpha. \] In this inversion formula, the double integral on the right-hand side is an ordinary improper Riemann integral that equals, rather than merely approximates, the average of f over the disc of radius c centered at (x,y). The novel contribution of this paper is to exhibit an explicit expression for the convolution kernel \(G_ c\) and utilize it for didactic purposes.