Perimeter and Area of a Rectangle

What defines a rectangle, how to find its perimeter with P = 2(l + w) and its area with A = l x w, and why a square is a special case.

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Rectangles are everywhere: books, doors, windows, tables, screens, sports pitches. Two measurements describe one completely — the distance around it and the amount of surface inside it.

Concept What makes a rectangle
length width
Four sides, with opposite sides equal.
All four angles are right angles.
The longer side is the length.
The shorter side is the width.
Concept Perimeter — the distance around

The perimeter is what you would walk if you followed all four sides. Adding them one by one always works:

P = l + w + l + w
Since each measurement appears twice, this shortens to P = 2(l + w) .

Both forms give the same number. The second is quicker, and it is the distributive property doing the work.

Concept Area — the surface inside

Area = length × width

Picture the rectangle covered in unit squares. The length tells you how many squares fit in a row and the width how many rows there are, so multiplying gives the total number of squares.

Example A 3 cm by 2 cm rectangle

Find the perimeter and the area.

Perimeter by adding: 3 + 2 + 3 + 2 = 10 cm.
Perimeter by formula: 2(3 + 2) = 2 \times 5 = 10 cm.
Area: 3 \times 2 = 6 cm².
⟹ perimeter 10 cm, area 6 cm²

Note the units. A perimeter is a length and is measured in cm; an area covers a surface and is measured in cm².

Example A larger rectangle

A room measures 8 m by 5 m.

Perimeter: 2(8 + 5) = 2 \times 13 = 26 m.
Area: 8 \times 5 = 40 m².
⟹ 26 m of skirting board, 40 m² of flooring

This is the practical difference between the two: perimeter buys what goes around the edge, area buys what covers the middle.

Example The square, a special case

A square is a rectangle whose length and width are equal, so the formulas simplify. Take a square of side 4 cm:

Perimeter: 2(4 + 4) = 16 cm, or simply 4 \times 4 .
Area: 4 \times 4 = 16 cm².
⟹ perimeter 16 cm, area 16 cm²

The two happen to share the number 16 here, but they are different quantities with different units. That coincidence occurs only for a side of 4.

Note Mistakes to avoid
Adding length and width once instead of twice for the perimeter.
Multiplying when the perimeter is wanted, or adding when the area is.
Forgetting the bracket: 2 \times l + w is not 2(l + w) .
Writing an area in cm rather than cm².
Assuming every four-sided shape is a rectangle; the angles must be right angles.
Summary
  1. A rectangle has opposite sides equal and four right angles.
  2. Perimeter is the distance around: P = 2(l + w).
  3. Area is the surface inside: A = l × w.
  4. Perimeter uses linear units; area uses square units.
  5. A square is a rectangle with equal sides, so both formulas still apply.
231. Arithmetic Sequences and Series
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