What Is a Rectangle?
A rectangle is a quadrilateral (a four-sided shape) in which all four interior angles are right angles ($90^\circ$ each). The longer pair of sides is usually called the length ($l$) and the shorter pair the width ($w$), though either pair can be named first.
Because a rectangle has four right angles and opposite sides parallel, it is a special kind of parallelogram, so it inherits every parallelogram property and adds the right-angle condition on top. That inheritance is why a rectangle's diagonals bisect each other, a fact it shares with all parallelograms, while the equal-diagonals fact below is special to the rectangle.
Properties of a Rectangle: Sides
The sides of a rectangle follow a clean, two-length pattern.
Opposite sides are equal. $AB = DC$ (the lengths) and $AD = BC$ (the widths).
Opposite sides are parallel. Each pair runs in the same direction and never meets.
Four sides in total, with two distinct side lengths in general.
A square is the special case where all four sides are equal, so every square is a rectangle, though most rectangles are not squares, a one-way relationship worth holding onto.
Properties of a Rectangle: Angles
The angles are the rectangle's defining feature.
All four interior angles are $90^\circ$. This is what makes the corners "square".
The interior angles sum to $360^\circ$, as in every quadrilateral: $4 \times 90^\circ = 360^\circ$.
Adjacent angles are supplementary, adding to $180^\circ$, which any two right angles do automatically.
A common real question is, "Is a rectangle a parallelogram?" Yes. A rectangle meets the parallelogram condition (two pairs of parallel sides) and then restricts every angle to $90^\circ$. So a rectangle is a parallelogram with square corners, which is exactly why it carries all the parallelogram diagonal facts plus one of its own.
Properties of a Rectangle: Diagonals
The diagonals are where the rectangle quietly earns its reputation for being "true".
The diagonals are equal in length. $AC = BD$. This is the property that separates a rectangle from a general parallelogram, whose diagonals are usually unequal.
The diagonals bisect each other. They cross at the centre point $O$, each cut into two equal halves, so $AO = OC$ and $BO = OD$.
The diagonals are generally not perpendicular. They meet at right angles only in the special case of a square.
The equal-diagonals property is no coincidence. Each diagonal is the hypotenuse of a right triangle whose legs are the length and width, and since both triangles share the same two legs, both hypotenuses come out the same. That is also the doorway to the diagonal-length formula below.
The Area, Perimeter, and Diagonal Formulas
A frequent question is, "What are the rectangle formulas and where do they come from?" All three follow directly from the sides and the right angles.
Area. The area is length times width:
$$A = l \times w.$$
This is the count of unit squares that tile the rectangle: $l$ squares across, $w$ rows of them, so $l \times w$ squares in all. Area is in square units.
Perimeter. The perimeter is the total distance around. With two lengths and two widths:
$$P = 2l + 2w = 2(l + w).$$
Diagonal. Each diagonal is the hypotenuse of a right triangle with legs $l$ and $w$. By the Pythagorean theorem:
$$d = \sqrt{l^2 + w^2}.$$
Symbol | Meaning | Units |
|---|---|---|
$l$ | Length, the longer side | length units (cm, m) |
$w$ | Width, the shorter side | length units (cm, m) |
$A$ | Area enclosed | square units ($\text{cm}^2$) |
$P$ | Perimeter, distance around | length units (cm) |
$d$ | Diagonal length | length units (cm) |
Is a Square a Rectangle?
Yes, a square is a rectangle. A square satisfies every rectangle property, four right angles and equal opposite sides, and simply adds the extra condition that all four sides are equal. So every square is a rectangle, but not every rectangle is a square. The relationship runs one way, the same nesting you saw with the rectangle sitting inside the parallelogram family. For the full treatment, see the sibling article on whether a square is a rectangle.
Examples of the Properties of a Rectangle
With the sides, angles, diagonals, and formulas in place, here is the rectangle doing real work. The problems build from a single area up to working backward from a known perimeter.
Example 1 - A rectangle has length $12$ cm and width $5$ cm. Find its area
$A = l \times w = 12 \times 5 = 60$.
Final answer: $60 \ \text{cm}^2$.
Example 2 - A rectangular field is $40$ m long and $30$ m wide. Find the length of fencing needed to go around it once
A tempting first move is to multiply the sides, the way you would for area: $40 \times 30 = 1200$, and call it the fencing. Check that against the question. Multiplying length by width gives the area the field covers, in square metres, not the distance around it. Fencing follows the boundary, so it needs the perimeter.
Done correctly: $P = 2(l + w) = 2(40 + 30) = 2 \times 70 = 140$ m.
Final answer: $140 \ \text{m}$ of fencing.
Example 3 - A rectangle has length $8$ cm and width $6$ cm. Find the length of its diagonal
The diagonal is the hypotenuse of a right triangle with legs $8$ and $6$: $d = \sqrt{l^2 + w^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10$ cm.
Final answer: $d = 10 \ \text{cm}$.
Example 4 - A rectangle has area $99 \ \text{m}^2$ and length $11$ m. Find its width
Work backward from $A = l \times w$: $99 = 11 \times w$, so $w = \dfrac{99}{11} = 9$ m.
Final answer: $w = 9 \ \text{m}$.
Example 5 - A rectangle has perimeter $28$ cm and length $8$ cm. Find its width
From $P = 2(l + w)$: $28 = 2(8 + w)$, so $8 + w = 14$, giving $w = 6$ cm.
Final answer: $w = 6 \ \text{cm}$.
Example 6 - The diagonals of a rectangle are equal. One diagonal measures $13$ cm. A student is told the rectangle's width is $5$ cm. Find its length
Because both diagonals are equal and each is the hypotenuse of a right triangle with legs $l$ and $w$: $d^2 = l^2 + w^2$, so $13^2 = l^2 + 5^2$, giving $169 = l^2 + 25$, so $l^2 = 144$ and $l = 12$ cm.
Final answer: $l = 12 \ \text{cm}$.
Why the Properties of a Rectangle Matter Beyond the Page
The rectangle is the workhorse shape of the built world, and its properties are the reason it is trusted everywhere right angles need to hold.
Squaring a structure. Builders check that a frame is a true rectangle by measuring both diagonals. If the diagonals are equal, the corners are square; if not, the frame is a leaning parallelogram. This is the equal-diagonals property doing quality control on a construction site.
Screens, pages, and packing. Phone screens, sheets of paper, shipping boxes, and floor tiles are rectangular because right angles tile a plane with no gaps and stack without wasted space.
Coordinate geometry. Area as $l \times w$ underpins how we measure regions on a grid, and the diagonal formula is the distance formula in disguise, the same Pythagorean step you will use to measure between any two points on a coordinate plane.
Strength from the square corner. A rectangular beam resists bending predictably because its cross-section's right angles distribute load evenly, which is why structural lumber and steel are milled rectangular rather than into odd quadrilaterals.
Where Students Trip Up on the Properties of a Rectangle
Mistake 1: Confusing area with perimeter
Where it slips in: A problem asks for fencing, tiling, or "the distance around", and the student multiplies the sides.
Don't do this: Use $l \times w$ when the question wants the boundary length.
The correct way: Area ($l \times w$) is the space inside, in square units; perimeter ($2(l+w)$) is the distance around, in length units. Read the question for the cue word. The rusher who grabs the two numbers and multiplies skips this read every time.
Mistake 2: Assuming the diagonals are perpendicular
Where it slips in: A student treats a rectangle like a square and claims its diagonals cross at $90^\circ$.
Don't do this: Set the diagonal crossing to a right angle.
The correct way: A rectangle's diagonals are equal and bisect each other, but they are perpendicular only in a square. The memorizer who blends square facts with rectangle facts loses marks here; only the special case has perpendicular diagonals.
Mistake 3: Forgetting the square root in the diagonal formula
Where it slips in: Computing the diagonal, the student stops at $l^2 + w^2$.
Don't do this: Report the diagonal as $l^2 + w^2$ without taking the root.
The correct way: $d = \sqrt{l^2 + w^2}$. The sum of the squares gives $d^2$, not $d$; the square root is the final step. The second-guesser who knows the Pythagorean theorem but rushes the last line drops the root here.
Conclusion
The defining property of a rectangle is its four right angles, from which every other property follows.
Opposite sides are equal and parallel, making a rectangle a special parallelogram.
The diagonals are equal in length and bisect each other, the equal-length part being unique to the rectangle within the parallelogram family.
Area is $l \times w$, perimeter is $2(l+w)$, and the diagonal is $\sqrt{l^2 + w^2}$ by the Pythagorean theorem.
Every square is a rectangle, but not every rectangle is a square.
Work Through These Problems to Test Your Understanding
A rectangle has length $15$ cm and width $8$ cm. Find its area and its perimeter.
A rectangle has length $24$ cm and width $7$ cm. Find its diagonal.
A rectangle has area $84 \ \text{cm}^2$ and width $7$ cm. Find its length.
Answer to Question 1: area $120 \ \text{cm}^2$, perimeter $46$ cm. Answer to Question 2: $d = 25$ cm. Answer to Question 3: $l = 12$ cm. If your two answers in Question 1 came out the same, check that you used $l \times w$ for area and $2(l+w)$ for perimeter, not the same operation twice.
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