How the calculator works
It runs your plan one year at a time. While you work, your savings grow at your return minus inflation and your yearly saving is added. From your retirement age, each year's gap (spending minus Social Security and pensions) is taken out and the rest keeps growing at your retirement return.
"You'll need" is the lump sum that would pay that gap every year until your plan-to age, given your retirement return. If your projection beats it, the curve stays above zero to the end. If it falls short, the curve hits the ground and the age where it does is the headline number.
Why a few years later changes so much
Working longer pulls three levers at once: more years of saving, more years of growth, and fewer years to fund. Use the "+2 / +3 / +5 years" buttons on the chart to see it. Delaying Social Security raises your other income too, which the Social Security calculator shows by claiming age.
The 25× shortcut
The quick version of this sum is to multiply the yearly gap by 25. It comes from the "4% rule" in the 1998 Trinity study of US market history, where a 4% first-year withdrawal, raised with inflation, lasted 30 years in most historical periods. Test your own rate with the withdrawal calculator, or stress-test it with the Monte Carlo simulation.