The color code is a topological quantum error-correcting code with a variety of computationally-valuable and fault-tolerant logical gates. Its two-dimensional version, the triangular color code, may soon be realized with currently available superconducting hardware limited by qubit connectivity. To guide the experimental effort, we thus study the storage threshold of the triangular color code against the circuit-level depolarizing noise. First, we adapt the Restriction Decoder to the setting of the triangular color code and to phenomenological noise. Then, we propose a fault-tolerant implementation of the stabilizer measurement circuits, which incorporates flag qubits. We show how information from flag qubits can be use with the Restriction Decoder to maintain the effective distance of the code. We numerically estimate the threshold of the triangular color code to be 0.2%, which is competitive with the thresholds of other topological quantum codes. Lastly, we prove that using 1-flag stabilizer measurement circuits are enough to recover the full distance of the code.