In a data hungry world, approximate computing has emerged as one of the solutions to create higher energy efficiency and faster systems, while providing application tailored quality. In this paper, we propose ApproxLP, an Approximate Multiplier based on Linear Planes. We introduce an iterative method for approximating the product of two operands using fitted linear functions with two inputs, referred to as linear planes. The linearization of multiplication allows multiplication operations to be completely replaced with weighted addition. The proposed technique is used to find the significand of the product of two floating point numbers, decreasing the high energy cost of floating point arithmetic. Our method fully exploits the trade-off between accuracy and energy consumption by offering various degrees of approximation at different energy costs. As the level of approximation increases, the approximated product asymptotically approaches the exact product in an iterative manner. The performance of ApproxLP is evaluated over a range of multimedia and machine learning applications. A GPU enhanced by ApproxLP yields significant energy-delay product (EDP) improvement. For multimedia, neural network, and hyperdimensional computing applications, ApproxLP offers on average 2.4×, 2.7×, and 4.3× EDP improvement respectively with sufficient computational quality for the application. ApproxLP also provides up to 4.5× EDP improvement and has 2.3× lower chip area than other state-of-the-art approximate multipliers.