When a box rests on a round sheet of wood on the ground, it exerts an average pressure p on the wood. if the wood is replaced by a sheet that has half the diameter of the original piece, what is the new average pressure? i think the answer is 4p but i don't know why.

Pressure = Force ÷ Area

In this case, since the area is circular, we can rewrite the equation as:

Pressure = Force ÷ pi*r^2

Decreasing the diameter by half is the same as decreasing the radius by half.

We can replace all constants with 1 to compare the relationship between the two scenarios.

Original scenario with original area wood:

Pressure = 1 ÷ 1^2
Pressure = 1

Scenario with 0.5 radius of original scenario:
Pressure = 1 ÷ (1/2)^2
Pressure = 1 ÷ 1/4
Pressure = 4

This confirms that your guess of 4x the pressure is correct.

So as can be seen by the relationship, pressure for circular area is inversely proportional to the radius squared.