Area Calculator

Area is flat, and the unit carries a square

Area answers one question: how much flat surface does this shape cover. A rug, a garden bed, a window pane and a parking spot all have an area, and all of them are measured in square units because two lengths are multiplied together.

Because two lengths go in, the unit gets multiplied as well. A foot is 12 inches, so a square foot is 12 x 12 = 144 square inches. A room that measures 12 feet by 9.5 feet covers 114 square feet, which is 114 x 144 = 16,416 square inches, or 114 x 0.09290304 = 10.59 square meters.

That single step explains most wrong answers. People convert feet to meters with 0.3048 and keep the same number for area. Do that here and 114 x 0.3048 = 34.75, an answer that is 3.3 times too large because the factor was applied once instead of twice. The square meter is the SI unit of area, defined as the area of a square one meter on each side in the SI brochure published by the BIPM, and NIST lists the whole ladder from square millimeter up to square kilometer on its SI units for area page.

Formulas for the shapes people measure

Five formulas cover almost every household or homework question. Everything else is these five cut up and added back together.

• Rectangle: area = length x width, perimeter = 2 x (length + width).
• Square: area = side x side, which is the rectangle formula with both sides equal.
• Triangle: area = 0.5 x base x height. The 0.5 is there because a triangle fills half of the rectangle that would surround it.
• Circle: area = pi x radius x radius, with pi at 3.14159 for hand work. Circumference = 2 x pi x radius.
• Trapezoid: area = ((side a + side b) / 2) x height, where a and b are the two parallel sides.
• Parallelogram: area = base x height, with no 0.5 and no correction for the slant.

Every one of those uses height, and height has a strict meaning: the straight line distance between the parallel sides, measured at a right angle. In a parallelogram a 6 foot slanted side on a 5 foot tall shape gives an area from 5, not from 6. In a triangle the same rule applies, so a base of 16 inches and a height of 9 inches gives 0.5 x 16 x 9 = 72 square inches whatever the other two sides measure.

Three worked examples with the arithmetic shown

A 12 by 9.5 foot room. 12 x 9.5 = 114 square feet. Carpet is sold by the square yard, so 114 / 9 = 12.67 square yards. Order 13 to allow for the edge cuts, since carpet rolls come in fixed widths and the installer will trim.

A round trampoline mat, 14 feet across. The radius is half the diameter, 7 feet. Area = 3.14159 x 7 x 7 = 153.94 square feet. The steel frame around it needs the circumference: 2 x 3.14159 x 7 = 43.98 feet of tubing, and a safety pad of the same shape therefore needs about 44 linear feet.

A garden bed that is wider at one end. One parallel side is 10 feet, the other is 14 feet, and the distance between them is 6 feet. Average the parallel sides: (10 + 14) / 2 = 12. Then 12 x 6 = 72 square feet. At a 3 inch depth that is 72 x 0.25 = 18 cubic feet of soil, or about 0.67 cubic yards.

Table: formula, example and answer for six shapes

The same table works in any length unit. Swap feet for meters and the answers become square meters with the identical arithmetic, which is exactly why the unit label matters more than the digits.

Area units convert by the square of the length factor

These are the conversions worth memorizing, and each one is exact or close enough for any home project:

A quarter-acre lot is 43,560 / 4 = 10,890 square feet, which is 10,890 x 0.09290304 = 1,011.71 square meters. The NIST guide to the SI, SP 811, prints these factors with the exact digit strings the definitions imply, and it is the reference to use when a client, a permit office or a teacher asks where a number came from.

One unit that catches people out is the square inch. A 3,000 square inch countertop sounds large and is only 3,000 / 144 = 20.83 square feet once converted, which is a small kitchen island.

Composite shapes: add the parts and subtract the holes

Real floors and yards are rarely one clean shape. Split them into rectangles, then subtract anything that is not being covered.

An L-shaped deck is 20 feet by 14 feet with an 8 by 6 foot notch cut out for the steps. The full rectangle is 20 x 14 = 280 square feet, the notch is 8 x 6 = 48 square feet, and 280 - 48 = 232 square feet of decking. Add 10 percent for waste and the order becomes 255 square feet, which is 23 square feet of extra boards for the diagonal cuts at the corner.

A round patio with a square fire pit in the middle works the other way. A 14 foot circle is 3.14159 x 7 x 7 = 153.94 square feet, the 3 by 3 foot pit cover is 9 square feet, and 153.94 - 9 = 144.94 square feet of pavers remain.

Two habits prevent the usual errors. Convert everything to one unit before adding, because a part measured in inches will silently disappear into a total in feet. And keep the subtraction in the same unit as the addition, so the hole and the surface cancel cleanly.

Scale factor: double the sides and the area goes up four times

Area grows with the square of any linear scale change. A 12 by 9 foot deck is 12 x 9 = 108 square feet. Build the same deck at twice the dimensions, 24 by 18, and it is 24 x 18 = 432 square feet: 432 / 108 = 4, so twice the width and twice the length bought four times the surface.

This is the honest way to compare pizzas and lots. A 16 inch pizza has 3.14159 x 8 x 8 = 201.06 square inches of surface and a 12 inch pizza has 3.14159 x 6 x 6 = 113.1 square inches, so 201.06 / 113.1 = 1.78: the larger pie holds about 78 percent more pizza, not 33 percent more even though the diameter is only a third larger. The same arithmetic explains why a 40 by 60 foot lot is not twice a 20 by 30 foot lot, it is four times the ground.

Halve the dimensions and the area drops to a quarter, which is the version homeowners meet when a plan drawn at half size looks affordable and the material list says otherwise.

Where this page stops

This tool covers flat shapes: it takes any length unit, returns the area in square inches, square feet, square yards, square meters and acres, and gives the perimeter when the shape has a fixed one. It does not know about walls, rooms, openings or 3D surfaces.

For paint, primer or drywall on real walls, the wall area calculator subtracts doors and windows per wall and totals several walls, which this page will not do. For a box, cylinder, cone or sphere, the surface area calculator counts every outside face of a solid. For a floor plan with several rooms measured in feet and inches, the square footage calculator adds the rooms and handles the inches to feet step. Reach for this page when the question is the area of one shape and you want to see it in several square units at once, or when you need the perimeter beside the area to price trim, fencing or edging.

If you only need the third side or an angle of a triangle, the triangle calculator solves the triangle from three known parts, and the right triangle calculator handles the 90 degree case with the Pythagorean theorem.