The shockwave generated by a body shaped like an acute cone is a (somewhat less acute) cone with the same apex. Inside the shock wave, the air gradually turns around to move parallel to the body, like so [0]. Each streamline that enters the shock structure keeps turning around, so each bit of air keeps feeling a force even after it passes through the original cone-shaped shock. In the case of a cone-shaped body, the interaction happens to create a single "straight" wave, but it's still generated by the entire surface of the cone, not just the sharp point of it.
As a rule of thumb, the magnitude of the sonic boom is proportional to the weight of the plane. That is because most of the interaction between the plane and the air is from the wing, it needs to continuously deflect down enough air to generate a lifting force. So the total boom has one contribution from the cross-section of the fuselage, and another from the wings, and the latter is determined by the weight.
[0] https://www.researchgate.net/profile/Richard-Benay/publicati...