we interpret "pyramid has a surface area of 258cm." to mean "pyramid has a surface area of 258cm^2."

let 'PI' represent the numeric constant approximated by 3.14, 'b' the length of the pyramid base, 'r' the radius of the cone, 'h' the slant height of the pyramid and 's' the slant height of the cone

total surface area for the pyramid:

b^2 + bh

total surface area for the cone:

PI*r*s + PI*r^2

PI*4s + 16*PI

we know that the surface areas of the cone and pyramid are equal and that the surface area of the pyramid is 258cm^2, so we can solve for 's', the slant height of the cone:

258 = PI*4s + 16*PI

solving for 's' we have s=16.530

the height of the cone is approximately 16.5 cm