The question you opened
A patio 25 feet on each side
A square patio measures 25 feet along each side.
How many square feet is the patio?
625 square feet
For a number ending in 5, take the front digit 2, multiply by the next one up, 3, to get 6, then write 25 after it: 625.
A square patio 25 feet on a side covers 625 square feet. There is a rule for squaring any number that ends in 5, and it takes about two seconds once you have seen it. Take the part of the number in front of the 5. Here that is 2. Multiply it by the next whole number up, which is 3.
That gives 6. Now write 25 on the end. The answer is 625. It works every time. For 35, take 3 times 4 to get 12, write 25, giving 1225. For 65, take 6 times 7 to get 42, so 4225. For 95, take 9 times 10 to get 90, so 9025. For 125, take 12 times 13 to get 156, so 15625.
The rule is not magic. Any number ending in 5 can be written as ten times a whole number plus 5, say 10n plus 5. Square that and you get 100n squared plus 100n plus 25, which is 100 times n times n plus one, plus 25. The 100 times n times n plus one is exactly the front part of the answer, and the 25 sits behind it.
That is the whole proof, and it explains why the last two digits are always 25. Areas are where this pays off in daily life. Flooring, turf, paint, decking and tiles are all sold by area, and quotes tend to arrive per square foot or per square metre. Being able to square a side length in your head lets you sanity check a quote while the person is still talking.
One more thing worth carrying. Area grows with the square of the length, so a patio 30 feet on a side is 900 square feet, not a little more than 625. Adding a fifth to the side adds nearly half again to the area. That gap is the reason a slightly bigger room costs so much more to floor.
Technique: Squares ending in five