Rectangle

Rectangle

A box shape on a plane. Formally, a rectangle is a quadrilateral with four congruent angles (all 90°).

Rectangle with labeled sides: height (vertical) and width (horizontal). Formula shown: Area = (height)(width)

Key Formula

A = l \times w \qquad P = 2l + 2w \qquad d = \sqrt{l^2 + w^2}

Where:

$A$ = Area of the rectangle
$P$ = Perimeter of the rectangle
$d$ = Length of a diagonal of the rectangle
$l$ = Length (the longer side, by convention)
$w$ = Width (the shorter side, by convention)

Worked Example

Problem: A rectangle has a length of 12 cm and a width of 5 cm. Find its area, perimeter, and diagonal length.

Step 1: Find the area by multiplying length times width.

A = l \times w = 12 \times 5 = 60 \text{ cm}^2

Step 2: Find the perimeter by adding all four sides. Since opposite sides are equal, use the formula with 2l + 2w.

P = 2l + 2w = 2(12) + 2(5) = 24 + 10 = 34 \text{ cm}

Step 3: Find the diagonal using the Pythagorean theorem. A diagonal splits the rectangle into two right triangles.

d = \sqrt{l^2 + w^2} = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \text{ cm}

Answer: The rectangle has an area of 60 cm², a perimeter of 34 cm, and a diagonal of 13 cm.

Another Example

This example works backward from area to find a missing dimension, then uses it to compute other properties. It also shows a case where the diagonal is not a whole number.

Problem: The area of a rectangle is 96 m² and its length is 12 m. Find the width, perimeter, and diagonal.

Step 1: Rearrange the area formula to solve for the unknown width.

w = \frac{A}{l} = \frac{96}{12} = 8 \text{ m}

Step 2: Now that you know both dimensions, compute the perimeter.

P = 2(12) + 2(8) = 24 + 16 = 40 \text{ m}

Step 3: Find the diagonal length.

d = \sqrt{12^2 + 8^2} = \sqrt{144 + 64} = \sqrt{208} = 4\sqrt{13} \approx 14.42 \text{ m}

Answer: The width is 8 m, the perimeter is 40 m, and the diagonal is

4\sqrt{13} \approx 14.42

Frequently Asked Questions

Is a square a rectangle?

Yes. A square has four right angles and opposite sides that are parallel and equal, so it satisfies every requirement of a rectangle. A square is simply a special rectangle where all four sides happen to be the same length. Every square is a rectangle, but not every rectangle is a square.

What is the difference between a rectangle and a parallelogram?

Every rectangle is a parallelogram (both have two pairs of parallel sides), but not every parallelogram is a rectangle. A parallelogram only requires opposite angles to be equal — they can be any measure. A rectangle adds the constraint that all four angles must be exactly 90°.

How do you find the diagonal of a rectangle?

Use the Pythagorean theorem. A diagonal connects two opposite corners and forms a right triangle with the length and width. So

d = \sqrt{l^2 + w^2}

. Both diagonals of a rectangle are equal in length.

Rectangle vs. Square

	Rectangle	Square
Angles	All four angles are 90°	All four angles are 90°
Sides	Opposite sides equal; adjacent sides can differ	All four sides equal
Area formula	A = l × w	A = s² (since l = w = s)
Diagonals	Equal in length; bisect each other	Equal in length; bisect each other at 90°
Classification	A special parallelogram	A special rectangle (and a special rhombus)

Why It Matters

Rectangles are one of the most common shapes you will encounter in geometry courses and standardized tests. Area and perimeter problems involving rectangles appear in algebra, coordinate geometry, and real-world applications like calculating floor space, screen dimensions, or material needed for construction. Understanding the rectangle also helps you grasp the classification hierarchy of quadrilaterals — parallelogram → rectangle → square — which is a frequent exam topic.

Common Mistakes

Mistake: Confusing area and perimeter formulas — for example, computing 2(l + w) when area is asked, or l × w when perimeter is asked.

Correction: Remember that area measures the space inside (square units) and uses multiplication: A = l × w. Perimeter measures the distance around the outside (linear units) and uses addition: P = 2l + 2w.

Mistake: Assuming a rectangle must have two different side lengths and therefore a square is not a rectangle.

Correction: The definition of a rectangle only requires four right angles. It does not forbid equal side lengths. A square meets all the conditions, so it is a rectangle — just a special one.

Related Terms

Area of a Rectangle — Formula for the interior space of a rectangle
Square — A rectangle with all four sides equal
Parallelogram — Parent category; rectangle is a special case
Quadrilateral — Any four-sided polygon; broader category
Congruent — Describes the equal angles and opposite sides
Angle — All four interior angles are 90°
Plane — The flat surface a rectangle lies on
Polygon — General term for closed shapes with straight sides