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For calculating pulse width in a standard IC 555 astable and monostable configurations we use the following formulas respectively:
$$T(\text{high}) \approx 0.693(R_1 + R_2)C_1$$
$$T = 1.1RC $$
And for the frequency we use the formula:
$$F = 1.44/((R_1+2R_2)C_1$$
How were the constants 0.693, 1.1, and 1.44 derived in the respective calculations?
The precise numbers are:
It's a good exercise for you to figure out how these numbers arise from the analysis of the circuit operation.
The key facts about the 555 are:
In monostable operation, the timing capacitor charges through a resistance from 0V to 2/3 Vcc.
In astable operation, the timing capacitor voltage cycles between 1/3 Vcc and 2/3 Vcc.
Output voltage of an RC circuit at time \$t\$ is $$V=V_{\text{max}}e^{-t/(RC)}$$ (negative based on charging or discharging) thus $$t=RC\ln\left(\frac{V}{V_{\text{max}}}\right)$$
For an astable multivibrator, charging and discharging happens between 2/3 and 1/3 Vcc, respectively. As initially it starts from zero charge, the on time (time taken to reach 2/3 Vcc - Time taken to reach 1/3 Vcc) i.e. is $$t = RC(\ln(2/3)-\ln(1/3)) = 0.693 RC$$
