Rectangle Area Calculator
Same perimeter, wildly different areas.
Work out Rectangle Area. Same perimeter, wildly different areas. Shows the working, not just the answer.
Area (square units)
12
Perimeter 14 · diagonal 5
Among all rectangles with a given perimeter, the square encloses the largest area — which is why the square of equal area always has a shorter perimeter than the rectangle. It is the same principle that makes a circle the most efficient shape of all for enclosing area.
How the Rectangle Area Calculator works
Area, perimeter, diagonal and aspect ratio of a rectangle, with the side of an equal-area square alongside. Perimeter and area are independent, and the gap between them is largest for long thin shapes.
Also known as: area of a rectangle · length times width calculator · room area in square metres · rectangle diagonal calculator
Perimeter and area are independent
A 1 × 9 rectangle and a 5 × 5 square both have a perimeter of 20, and their areas are 9 and 25. Knowing one tells you almost nothing about the other, which is less intuitive than it should be.
Among rectangles of fixed perimeter, the square encloses the most area, and the more elongated the shape the worse it does. It is a special case of the isoperimetric principle, where the circle is the overall winner.
The practical consequence appears in heat loss and in fencing. A compact building loses less heat than a sprawling one of equal floor area, and a square field needs less fence than a long thin one enclosing the same land.
Diagonals, aspect ratios and screens
The diagonal is Pythagoras applied to the sides, and a 3 by 4 rectangle famously gives exactly 5. It is why the 3-4-5 triangle is used to check a corner is square on site.
Screens are measured diagonally because it gives the largest number for a given panel. It also means two screens of the same diagonal have different areas if their aspect ratios differ — a 16:9 screen has noticeably less area than a 4:3 one.
The diagonal alone does not determine a rectangle, since infinitely many share it. It takes the aspect ratio as well, which is exactly how screen dimensions are worked back from a quoted diagonal.
Why A4 is the shape it is
The A series has an aspect ratio of the square root of two, chosen so that halving a sheet along its longer side produces the same proportions. A5 is exactly half an A4 with nothing cropped.
That property is unique to that ratio. Any other aspect ratio changes shape when halved, which is why every other paper standard needs separate proportions for each size.
A0 is defined as one square metre at that ratio, which fixes every other size in the series. It is why A4 paper weight is quoted in grams per square metre and why the arithmetic for postage works out so tidily.
Where to go next
The Rectangle Area question rarely arrives on its own. These are the ones that usually come with it:
- Circle Area Calculator — A circle fills 78.54% of its square, always.
- Perimeter Calculator — Three lengths only make a triangle if two beat the third.
- Polygon Area Calculator — Exterior angles always sum to 360°, whatever the shape.
- Percentage Calculator — Every common percentage question in one place.
Frequently asked questions
How do I calculate the area of a rectangle?
Length times width. Both must be in the same unit, and the result is in square units of that measure.
Do two rectangles with the same perimeter have the same area?
No, and the difference can be enormous. A 1 × 9 and a 5 × 5 both have a perimeter of 20 and areas of 9 and 25 — the square is always the largest.
How do I find the diagonal?
Pythagoras: the square root of length squared plus width squared. A 3 by 4 rectangle has a diagonal of exactly 5.
What is aspect ratio?
Length divided by width. A 16:9 screen has an aspect ratio of 1.778, and it is the number that decides whether an image fits without cropping.
How do I calculate an irregular room?
Break it into rectangles, calculate each and add them. For anything with a curved edge, approximate with rectangles or use the formula for that shape.
Why do TVs measure diagonally?
Because it gives the largest number for a given screen. It also means two screens of the same diagonal have different areas if their aspect ratios differ — a 16:9 screen has less area than a 4:3 one of the same diagonal.
What is the golden ratio in a rectangle?
An aspect ratio of about 1.618, where removing a square leaves a rectangle of the same proportions. Its aesthetic significance is much disputed and its mathematical properties are not.
Why is A4 paper the shape it is?
Its aspect ratio is the square root of two, which means halving it lengthways gives the same proportions. That is why A5 is exactly half an A4 with no cropping.
How do I find area from the diagonal?
You cannot without another dimension — infinitely many rectangles share a diagonal. With the aspect ratio as well it is determined, which is how screen sizes are worked out.
What is the area of a parallelogram?
Base times perpendicular height, not base times side. Shearing a rectangle into a parallelogram preserves the area while lengthening two of the sides.
How do I calculate square footage of a house?
Sum the rectangles of each room, and check which convention is being used — some measures include internal walls and some do not, which can change the total by several percent.
Why do rooms measure length before width?
Convention rather than mathematics; area is unaffected by the order. It matters for materials that come in fixed widths, where which dimension runs which way changes the waste.
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