What area really is
Area is how much flat surface something covers, measured in square units: square feet, square inches, square yards, square metres, acres. A square foot is a square one foot on each side, so a rectangle 12 ft by 8 ft holds 96 of them — twelve one way and eight the other, multiplied. The formula changes with the shape, but the idea never does: area is length times width, in one form or another, and the units are always squared, because two lengths are multiplied together.
The conversions are exact: 144 square inches in a square foot, 9 square feet in a square yard, 0.09290304 square metres in a square foot, and 43,560 square feet in an acre. A 96 sq ft rectangle is 13,824 sq in, 10.666667 sq yd, 8.918692 sq m and 0.002204 acres — the same area, in four other units. The area-calculator page works out all six shapes and all five units in one place, from your own measurements.
The worked example: 12 by 8
Take a rectangle 12 ft by 8 ft — a small bedroom, near enough. The first step is to multiply length by width: 12 × 8 = 96 sq ft. The second step is the conversions: 96 × 144 = 13,824 sq in; 96 ÷ 9 = 10.666667 sq yd; 96 × 0.09290304 = 8.918692 sq m; 96 ÷ 43,560 = 0.002204 acres.
The same two numbers give a different area for every shape. As a triangle, 12 and 8 make 48 sq ft — base times height over two. As the bases of a trapezoid with a height of 5 ft, they make 50 sq ft — the average base, 10, times the height. The shape decides the formula, not the numbers, so pick the shape first and the dimensions second. The area-calculator page takes all six shapes, so you never have to remember which formula goes where.
Circles: pi and the radius
A circle's area is pi times the radius squared: area = π × r². A circle of radius 5 ft is 3.141593 × 25 = 78.54 sq ft, unrounded 78.539816. The radius is half the diameter — measure across the whole circle and divide by two, or measure from the centre to the edge. The most common mistake is using the diameter as the radius: a circle 10 ft across has a radius of 5 ft, not 10 ft, and its area is 78.54 sq ft, not 314.16 sq ft.
Area grows with the square of the size: a circle of radius 10 ft is 314.16 sq ft, four times the radius-5 circle, because doubling the radius quadruples the area. The same is true of every shape — doubling one side doubles the area, but doubling both sides quadruples it. If your circle is twice as wide as you thought, its area is four times what you planned for.
Triangles and trapezoids
A triangle's area is base times height over two: a triangle of base 10 ft and height 6 ft is 10 × 6 ÷ 2 = 30 sq ft. The height is the straight-up distance from the base to the top corner, not the sloping side — measure it at a right angle to the base, or the answer is too big. Half of base times height is the whole rule, and it works for any triangle, right-angled or not.
A trapezoid's area is the average of its two parallel bases times the height: bases 8 ft and 12 ft with a height of 5 ft give (8 + 12) ÷ 2 × 5 = 50 sq ft. The two bases are the parallel sides — measure them first, then the height between them. A parallelogram is simpler still: base times height, 9 ft by 4 ft is 36 sq ft, the same as a 9 by 4 rectangle, slanted. The area-triangle-coordinates page works out a triangle's area from its three corner points, when base and height are hard to measure.
From area to square footage
Area becomes square footage when you buy flooring for it. A room 12 ft by 10 ft is 120 sq ft, but you buy 132 sq ft — 10% waste for the cuts around the edges and the mistakes — and at $5.00 a square foot the material cost is $660.00. The room, the amount to buy and the cost are three different numbers, and the square-footage-calculator page works out all three from your room's length and width.
Ten percent is the usual allowance for a plain room; a patterned floor or a room with many corners needs 15% or 20%. The smallest waste for 150 sq ft to buy is 25%: at 25% the amount to buy is 150.00 sq ft, and at 24% it is 148.80 sq ft. Measure wall to wall, in feet, at the longest and widest points — a room that narrows at one end is measured at its widest, and the waste covers the rest. The land-area-calculator page turns lot dimensions into acres, for the same job outdoors.
The checklist
Before you trust an area, check four things. First, the shape — the shape decides the formula, not the numbers, and the same two numbers give a different area for every shape. Second, the radius — half the diameter, not the whole of it, or the circle is four times too big. Third, the height — straight up from the base at a right angle, not the sloping side, or the triangle is too big. Fourth, the units — square feet, square yards and acres are different units, and 96 sq ft is 10.666667 sq yd, not 96 of them. Get those four right and the number is yours: 96 sq ft for a 12 by 8 rectangle, and 78.54 sq ft for a circle of radius 5 ft.