Find the area of a square, rectangle, circle or triangle.
Area is the space inside a two-dimensional shape, measured in square units. The core formulas are few: rectangle is length times width; triangle is half base times height; circle is pi times radius squared; trapezium is half the sum of the parallel sides times the height. Most irregular shapes are handled by splitting them into these and adding the parts, which is how flooring, painting and land measurement are done in practice.
Two errors account for most wrong answers. The first is using the slant side of a triangle instead of the perpendicular height — the height must be at right angles to the base, not along the sloping edge. The second is unit scaling. Area scales with the square of a linear dimension, so converting from metres to centimetres multiplies the area by 10,000, not 100. Doubling the sides of a room quadruples its floor area, which is also why a modestly larger tin or room needs disproportionately more material than the linear increase suggests.
The perpendicular distance from the base to the opposite vertex, at right angles to the base. It is not the length of the sloping side, and using the slope instead is the most common triangle area error.
Because area has two dimensions. One metre is 100 centimetres, but one square metre is 10,000 square centimetres, since both dimensions are converted. The factor is always the length factor squared.
Divide it into rectangles, triangles and circle segments, calculate each, then add them. Subtract any cut-outs. For genuinely irregular outlines, approximating with many narrow strips gives a workable estimate.
Because doubling both length and width multiplies the area by four. Area grows with the square of linear dimensions, so material requirements rise far faster than room size suggests.