# Properties of a Rectangle: Sides, Angles & Diagonals

TL;DR

A rectangle is a quadrilateral with four right angles, opposite sides equal and parallel, and diagonals that are equal in length and bisect each other. This article covers every property of a rectangle by sides, angles, and diagonals, derives the area l × w, perimeter 2(l + w), and diagonal \sqrt{l^2 + w^2} formulas, and works through six examples.

## What Is a Rectangle?

A **rectangle** is a quadrilateral (a four-sided shape) in which **all four interior angles are right angles** (90° 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 and AD = BC.
- **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.

## Properties of a Rectangle: Angles

The angles are the rectangle's defining feature.

- **All four interior angles are 90°.** This is what makes the corners "square".
- **The interior angles sum to 360°**, as in every quadrilateral: 4 × 90° = 360°.
- **Adjacent angles are supplementary**, adding to 180°.

A rectangle meets the parallelogram condition (two pairs of parallel sides) and then restricts every angle to 90°. So a rectangle is a parallelogram with square corners.

## 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 center point O, each cut into two equal halves.
- **The diagonals are generally not perpendicular.** They meet at right angles only in the special case of a square.

## The Area, Perimeter, and Diagonal Formulas

**Area.** The area is length times width:

A = l × w.

**Perimeter.** The perimeter is the total distance around:

P = 2l + 2w = 2(l + w).

**Diagonal.** Each diagonal is the hypotenuse of a right triangle with legs l and w:

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 (cm²) |
| 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.

## Examples of the Properties of a Rectangle

### Example 1 - A rectangle has length 12 cm and width 5 cm. Find its area.

A = l × w = 12 × 5 = 60.

Final answer: 60 cm².

### Example 2 - A rectangular field is 40 m long and 30 m wide. Find the length of fencing needed to go around it once.

P = 2(l + w) = 2(40 + 30) = 2 × 70 = 140 m.

Final answer: 140 m of fencing.

### Example 3 - A rectangle has length 8 cm and width 6 cm. Find the length of its diagonal.

d = \sqrt{l^2 + w^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10 cm.

Final answer: d = 10 cm.

### Example 4 - A rectangle has area 99 m² and length 11 m. Find its width.

w = \dfrac{99}{11} = 9 m.

Final answer: w = 9 m.

### Example 5 - A rectangle has perimeter 28 cm and length 8 cm. Find its width.

w = 6 cm.

Final answer: w = 6 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.

l = 12 cm.

Final answer: l = 12 cm.

## Conclusion

The defining **property of a rectangle** is its four right angles, from which every other property follows. Area is l × 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.