Square Area Calculator

Take a square of side 5. The area is 5 × 5 = 25. The picture behind that is a row of five unit squares repeated five times, which is why an area is measured in squared units: five metres by five metres gives twenty-five square metres. Backwards is less tidy. An area of 25 gives a side of exactly 5, because 25 is a perfect square. An area of 20 gives √20 = 2√5, about 4.4721, and no decimal is exactly right. That is the normal case: below ten thousand only 100 whole numbers are perfect squares, and below a million only 1,000. The gaps between them keep widening, so the larger the area, the less likely a tidy side. Now the tiling. A square room 10.1 metres across, tiled with one-metre tiles. By area: 10.1² = 102.01 square metres, divided by 1 square metre a tile, rounded up, is 103 tiles. By the floor: 10.1 ÷ 1 = 10.1, so you need 11 whole rows, and 11 whole columns, which is 11 × 11 = 121 tiles. The difference is 18 tiles, and it is not a rounding error. The area calculation assumes the 0.1-metre strips cut off the end of each row can be reassembled to cover the last row, and they cannot: they are 0.1 by 1, while the last row needs pieces 1 by 0.1 laid the other way, and the far corner needs a piece 0.1 by 0.1 cut from a whole tile of its own. Of the 121 tiles bought, 102.01 square metres of tile ends up on the floor and 18.99 is offcut — 15.7% waste. Smaller tiles waste less, because the overhang is only ever one tile wide: the same room tiled at 0.5 metres needs 21 rows of 21, which is 441 tiles covering 110.25 square metres, for 7.5% waste. When the tile divides the side exactly, the two counts agree and nothing is cut. A 10-metre room at one metre is 10 rows of 10, which is 100 tiles, and 100 is also what the area gives. Finally the scaling. Sides of 5, 10 and 15 give areas of 25, 100 and 225. Tripling the side makes the area nine times larger, not three, because two lengths are multiplied together and the scale factor applies to both.