This paper details a design optimization on a hardware accelerator for an on-board real-time SAR imaging system using the Backprojection algorithm, focusing on algorithm-specific approximations intended to reduce the overhead introduced by intensive functions such as the square root, sine, and cosine functions. The main novelty of this work is that new approximations were investigated so that the final image retains high quality regardless of the error in the approximation function. This paper revisits existing approximation methods, such as linear interpolation, polynomial approximation using the Vandermonde matrix, and Chebyshev polynomials, and compares their performance on the algorithm. Results demonstrate that it is possible to maintain the image quality with an SSIM above 0.99 with less 93% LUTs for the square root and less 88% LUTs for the sine and cosine functions using polynomial approximation with the Vandermonde matrix when compared to the HLS baseline functions.