Geometry
How To Calculate The Area Of A Rectangle
Rectangle area is the first formula most people learn and the one most often got wrong, because the units, the perimeter and the diagonal all get mixed into the same sentence.
Quick Answer
area = length x width
- A
- Area covered, in square units
- l
- Length of the longer side, in metres or feet
- w
- Width of the shorter side, in the same unit
Multiply the length by the width and you have the area in square units. An eight metre by five metre room covers 40 square metres. Convert both measurements to the same unit first, and remember the answer is square units, not plain metres.
What Is The Area Of A Rectangle?
The area of a rectangle is the amount of flat surface it covers, found by multiplying its length by its width. A rectangle eight metres long and five metres wide covers forty square metres, which is exactly how much carpet, paint or turf it would take to cover it.
The reason multiplication works is that you can tile the shape. Lay a row of five one metre squares along the short side and you have five square metres. Stack eight such rows and you have eight times five, or forty, squares. Area is really a count of unit squares, and the formula is a shortcut for counting them.
Area is measured in square units, and this is where most mistakes start. Multiplying eight metres by five metres gives forty square metres, written with a small two after the unit. Writing forty metres is wrong, because that is a length, not an area, and the two are not interchangeable.
Perimeter is the distance around the edge and is measured in plain units. For the same rectangle the perimeter is two times eight plus five, which is twenty six metres. Area and perimeter are different quantities, and a shape can have a large area but a small perimeter, as a long thin strip does.
A square is a rectangle whose sides are equal, so the same formula applies with length and width set to the same number. The area becomes the side squared. Every square is a rectangle, but not every rectangle is a square, which matters when a problem hands you one property but asks for another.
The diagonal of a rectangle comes from the Pythagorean theorem. It splits the shape into two right triangles, so the diagonal is the square root of length squared plus width squared. For an eight by five rectangle that is the square root of eighty nine, about 9.43 metres.
When units differ, convert before multiplying. A room measured as four metres by two hundred and fifty centimetres has to become either four metres by 2.5 metres or four hundred centimetres by two hundred and fifty centimetres first. Multiplying four by two hundred and fifty gives a thousand, which is ten times too small.
The same formula covers far more than floors. Wall area is height times width, so a wall four metres wide and 2.4 metres high covers 9.6 square metres. Subtract doors and windows, each of which is itself a rectangle, to get the paintable area.
For irregular spaces, split the floor plan into rectangles, work out each one and add the results. An L shaped room is two rectangles sharing an edge. The trap is double counting the overlap, so draw the split line before measuring anything.
Area scales with the square of length. Doubling both sides of a rectangle multiplies the area by four, not by two. That is why a room twice as wide and twice as long takes four times as much flooring, and why paint estimates go wrong when people scale by length alone.
Finally, area is always positive. If a measurement comes out negative, the sign is telling you about direction in a coordinate problem rather than about size. For physical lengths, take the absolute value before multiplying.
Formula
A = l x w
Length times width. Both measurements must be in the same unit, and the result is in that unit squared.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| A | Area | square metres | Flat surface covered by the rectangle. |
| l | Length | metres | The longer side. |
| w | Width | metres | The shorter side, in the same unit as the length. |
P = 2 x (l + w)
The distance around the edge. It is a plain length, not a square measure.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| P | Perimeter | metres | Total distance around the rectangle. |
| l | Length | metres | The longer side. |
| w | Width | metres | The shorter side. |
d = sqrt(l^2 + w^2)
Pythagoras applied to the right triangle the diagonal creates.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| d | Diagonal | metres | Corner to corner distance. |
| l | Length | metres | The longer side. |
| w | Width | metres | The shorter side. |
How To Calculate The Area Of A Rectangle
- 1
Measure the length and the width
Take the longer side as the length and the shorter as the width. It makes no difference to the area which one you call which, because multiplication is commutative, but it keeps the working readable.
- 2
Put both measurements in the same unit
Convert before multiplying. A length in metres and a width in centimetres cannot be multiplied directly, and doing so is the single most common source of answers that are out by a factor of a hundred.
- 3
Multiply length by width
Eight metres times five metres is forty. Nothing else is needed: no division, no addition, and no adjustment for the shape being long and thin.
- 4
Write the answer in square units
Forty square metres, not forty metres. The unit is the length unit multiplied by itself, so metres times metres gives square metres and feet times feet gives square feet.
- 5
Sanity check against the perimeter
The perimeter is two times the length plus the width, which for this rectangle is twenty six metres. If your perimeter is smaller than your area in the same unit, something has gone wrong with the units.
Examples
Example 1: A room eight metres by five
- Length
- 8 m
- Width
- 5 m
| Step | Calculation | Result |
|---|---|---|
| Perimeter | 2 x (8 + 5) | 26 |
| Diagonal | sqrt(8^2 + 5^2) | 9.43 |
| Area | 8 x 5 | 40 |
Result: An eight by five metre room covers 40 square metres of floor, has a perimeter of 26 metres of skirting and a corner to corner diagonal of about 9.43 metres.
Example 2: A wall four metres by 2.4 metres
- Length
- 4 m
- Width
- 2.4 m
| Step | Calculation | Result |
|---|---|---|
| Perimeter | 2 x (4 + 2.4) | 12.8 |
| Diagonal | sqrt(4^2 + 2.4^2) | 4.66 |
| Area | 4 x 2.4 | 9.6 |
Result: A wall four metres wide and 2.4 metres high covers 9.6 square metres, so a single litre of paint covering twelve square metres is more than enough for one coat.
Example 3: A garden bed twelve metres by three
- Length
- 12 m
- Width
- 3 m
| Step | Calculation | Result |
|---|---|---|
| Perimeter | 2 x (12 + 3) | 30 |
| Diagonal | sqrt(12^2 + 3^2) | 12.37 |
| Area | 12 x 3 | 36 |
Result: A long thin bed twelve metres by three has an area of 36 square metres but a perimeter of thirty metres, a reminder that the two measures move independently.
Calculator
Area
40
- Perimeter
- 26
- Diagonal
- 9.434
- Aspect ratio (long side / short side)
- 1.6
Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.
Prefer a full-width tool? Open the The Area Of A Rectangle calculator page.
Common Mistakes
Writing the answer in plain units
Forty square metres and forty metres are different quantities. The area unit is the length unit squared, and dropping the square makes the answer meaningless even when the arithmetic is right.
Multiplying measurements in different units
Eight metres by fifty centimetres is not four hundred. Convert to a single unit first, so the calculation becomes eight times 0.5, which is four square metres.
Adding the sides instead of multiplying
Adding gives the perimeter, not the area. It is an easy slip when a question mentions all four sides, and it produces an answer that looks plausible but is in the wrong dimension entirely.
Forgetting that a square uses the same formula
A square is a rectangle with equal sides, so the formula still applies. Some people reach for a separate rule and then get confused about which measurement to use.
Scaling area linearly
Doubling both sides multiplies the area by four. Ordering twice the flooring because the room is twice as long and twice as wide is one of the most expensive arithmetic errors in home improvement.
Double counting in L shaped rooms
Splitting a floor plan into rectangles works, but only if the rectangles do not overlap. Mark the split line on a sketch before measuring, or the shared strip gets counted twice.
Treating the diagonal as the width
The diagonal is longer than either side. Substituting it for the width inflates the area, and it is a natural slip when only the diagonal and one side have been measured.
FAQ
What is the formula for the area of a rectangle?
Area equals length multiplied by width. Both measurements must be in the same unit, and the answer comes out in that unit squared, so metres times metres gives square metres.
What is the difference between area and perimeter?
Area is the surface inside the shape, measured in square units. Perimeter is the distance around the outside, measured in plain units. A shape can have a large area and a small perimeter, or the reverse.
How do you find the area if you only know the diagonal?
You cannot, unless you also know one side or the angle. The diagonal plus one side gives the other side through Pythagoras, and then the area follows from length times width.
Why is area measured in square units?
Because you are counting how many unit squares fit inside. A square metre is one metre by one metre, so a shape forty of them in size has an area of forty square metres.
What happens if the length and width are in different units?
Convert one of them so both match before multiplying. Multiply centimetres by centimetres, or metres by metres, but never centimetres by metres.
Is a square a rectangle?
Yes. A square satisfies every condition of a rectangle, with the extra property that all four sides are equal. The rectangle area formula therefore works unchanged on a square.
References
- [1]Wikipedia, Rectangle — https://en.wikipedia.org/wiki/Rectangle
- [2]Wikipedia, Area — https://en.wikipedia.org/wiki/Area
- [3]Math is Fun, Area of a rectangle — https://www.mathsisfun.com/geometry/rectangle.html
- [4]Wikipedia, Pythagorean theorem — https://en.wikipedia.org/wiki/Pythagorean_theorem