Area of a Rectangle Calculator

Background.

The Quanta area of a rectangle calculator turns a length and a width into three answers in one keystroke: the area enclosed by the four sides, the perimeter that runs around them, and the diagonal that cuts from one corner to the opposite. The arithmetic is the most elementary in all of geometry — area equals length multiplied by width — but it is also the most relied-upon calculation in everyday life.

It is what tells you how much carpet to order for a 14-foot by 11-foot living room, how many litres of paint will cover the wall behind your sofa, how many tiles you need for a kitchen splashback, how much grass seed to spread on a back garden, how many square metres of cladding to ship to a construction site, and how much you should be paying per square foot for an apartment.

The perimeter sits underneath every estimate for fencing, skirting, picture-frame moulding, weatherstripping, and tape-edge trim. The diagonal underwrites every advertised screen size — when a manufacturer says "55-inch television" they mean the 55-inch diagonal of the 16:9 rectangle, not the width — and it is the number a carpenter measures from corner to corner to confirm that a freshly framed door opening is actually square (if the two diagonals match, the rectangle is true; if they differ, the frame is racked into a parallelogram).

The rectangle is the oldest measured shape in human civilization. Babylonian land surveyors recorded rectangular field areas on clay tablets in the third millennium BCE. Egyptian rope-stretchers used 3-4-5 right triangles to lay out rectangular foundations for temples and tombs along the Nile. Euclid formalised the area of a rectangle in Book I of the Elements around 300 BCE as the product of its sides, and proved (Proposition 35) that two parallelograms on the same base and between the same parallels are equal in area — the lemma that lets the rectangle's simple length-times-width rule extend to every parallelogram, triangle, trapezoid, and ultimately, through the method of exhaustion and modern integral calculus, to every region in the plane.

The formulas this page implements are length × width for area, 2 × (length + width) for perimeter, and √(length² + width²) for the diagonal; the last is a direct consequence of the Pythagorean theorem applied to the right triangle formed by two adjacent sides. The calculator rejects negative or zero side lengths because a rectangle with a side of zero degenerates into a line segment with no enclosed area. Use it for homework, for renovation estimates, for landscape planning, for screen-fit checks, or for the dozens of everyday situations — moving furniture, mounting a TV, ordering blinds, sizing a rug — where you need to turn two tape-measure readings into a useful number. Everything below the widget is the full explainer: what each formula means, where it comes from, how to apply it to real rooms and real projects, and the worked example that walks through a 10-by-5 rectangle from inputs to all three outputs.

What is area of a rectangle calculator?

A rectangle is a four-sided plane figure (a quadrilateral) in which all four interior angles are right angles (90°). A direct consequence of all four angles being right is that opposite sides are parallel and equal in length, which is why every rectangle can be fully described by just two numbers — a length and a width. The area of a rectangle is the number of unit squares that fit inside its boundary; for a rectangle whose sides are integers, you can literally count them, and the count always equals length × width. For non-integer sides the same formula extends by limit and continuity (this is the move Euclid makes in Elements Book VI and that modern calculus formalises with Riemann sums). The perimeter is the total distance around the outside, equal to the sum of all four sides — and because opposite sides are equal, that sum collapses to 2 × (length + width). The diagonal is the straight-line distance between any pair of opposite corners; because a diagonal splits the rectangle into two congruent right triangles whose legs are the length and the width, the diagonal is the hypotenuse, and by the Pythagorean theorem it equals √(length² + width²). A square is the special case of a rectangle in which length and width are equal — every square is a rectangle, but not every rectangle is a square. The rectangle is the foundational shape of measured space because almost every man-made surface — rooms, walls, floors, screens, sheets of paper, building lots, parking spaces, shipping pallets, photographs — is built or framed as a rectangle, and almost every measured estimate for material, coverage, or capacity reduces in the first step to a length-times-width product.

How to use this calculator.

1. 01Measure the length and the width of the rectangle in the same unit. If you are estimating a room, take the tape measure along one wall for the length, then along the perpendicular wall for the width. For a wall area, length is horizontal and width is vertical (the wall height).
2. 02Enter the length value into the first field. The calculator accepts decimals — so 14.5 feet or 4.32 metres both work — and you can type any positive number.
3. 03Enter the width value into the second field, in the same unit you used for the length. Mixing units (feet for length, inches for width) will produce a meaningless answer; convert first if you need to.
4. 04Read the primary output: Area. It is reported as "square units" — that is, the square of whatever unit you typed. Entered metres → square metres. Entered feet → square feet. Entered inches → square inches.
5. 05Read the two secondary outputs: Perimeter (the distance around the rectangle, in the original unit) and Diagonal (the corner-to-corner straight line, in the original unit). Use perimeter for trim, fencing, and edging estimates; use diagonal for screen-size, sheet-cut, and squareness checks.
6. 06For unit conversion downstream — for example, square metres to square feet for a flooring quote — multiply square metres by 10.7639 to get square feet, or divide square feet by 10.7639 to get square metres. The Quanta unit converter handles this in one step.

The formula.

Three formulas, all derived from the definition of a rectangle as a four-sided shape with four right angles and equal opposite sides. Area: A = l × w. The proof for integer side lengths is visual — divide the rectangle into a grid of l columns and w rows of unit squares, count them, and you get l × w. Euclid generalised this to all real-valued sides in Elements Book VI by the method of equal ratios, and modern measure theory formalises it through the product measure on ℝ². Perimeter: P = 2 × (l + w). Walk around the four sides once: you cover length, width, length, width — two of each — which sums to 2l + 2w, factored as 2(l + w). Diagonal: d = √(l² + w²). Draw a diagonal from any corner to the opposite corner. The diagonal cuts the rectangle into two right triangles, each with legs of length l and w. The diagonal itself is the hypotenuse of either of those triangles, so by the Pythagorean theorem (Euclid, Elements Book I, Proposition 47) the diagonal length is the square root of the sum of the squares of the two legs: d = √(l² + w²). A useful consequence: the two diagonals of any rectangle are equal in length and bisect each other at the centre of the rectangle. Carpenters and framers use this fact constantly — if you measure both diagonals of a door opening or window frame and they match, the opening is square; if they differ by more than a couple of millimetres, the frame is racked into a parallelogram and needs to be adjusted before the door or window is hung. The calculator returns all three values simultaneously because in practical work you almost always need more than one of them — an area for the material order, a perimeter for the trim, a diagonal for the layout check.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Euclid, Elements, Book I, Proposition 35 — parallelograms on the same base and between the same parallels are equal in area; the foundational lemma underlying the length × width formula for rectangle area (Heath translation, Cambridge University Press, 1908).
2. [2]Euclid, Elements, Book I, Proposition 47 — the Pythagorean theorem, which gives the diagonal of a rectangle as √(length² + width²).
3. [3]Coxeter, H. S. M. Introduction to Geometry, 2nd edition. Wiley, 1969 — classic reference geometry textbook covering rectangles, parallelograms, area, perimeter, and the Pythagorean theorem.
4. [4]Weisstein, Eric W. "Rectangle." MathWorld — A Wolfram Web Resource. Canonical definitions of rectangle area, perimeter, diagonal, and related identities.
5. [5]NIST Handbook of Mathematical Functions (Digital Library of Mathematical Functions), §1.6 — definitions of length, area, and volume in Euclidean geometry, including the rectangle as the fundamental unit of area measure.
6. [6]Joyce, David E. Euclid's Elements, Clark University — online edition of the Elements with commentary on Book I and Book VI propositions that establish rectangle area as the product of its sides.