Area and Perimeter of a Rectangle

Area and Perimeter of a Rectangle

How many tiles do you need to cover a room? How much fencing is needed for a garden? These questions are solved with two ideas: perimeter (the distance around the outside) and area (the amount of space inside). On this page we'll learn to calculate both for rectangles and squares.

Background and Basic Definitions

Perimeter = the sum of all sides. For a rectangle:

\[ P = 2 \times (a + b) \]

where \(a\) = length and \(b\) = width.

Area = the amount of surface inside the shape. For a rectangle:

\[ S = a \times b \]

Units of area: square centimetres (\(\text{cm}^2\)), square metres (\(\text{m}^2\)), etc.

A square is a rectangle where all sides are equal (\(a = b\)):

\[ P_{\text{square}} = 4a \qquad S_{\text{square}} = a^2 \]

MeasureFormula — RectangleFormula — Square
Perimeter\(2(a+b)\)\(4a\)
Area\(a \times b\)\(a^2\)
S = a · b b a length width
Rectangle: length, width, area and perimeter

Solution Steps

  1. Step 1 — Identify the shape: rectangle (length ≠ width in general) or square (all sides equal).
  2. Step 2 — Read what is being asked: perimeter (the boundary line) or area (the inside surface).
  3. Step 3 — Check the units: are all measurements in the same unit? If not, convert first (e.g., cm to m).
  4. Step 4 — Calculate using the formula: perimeter = \(2(a+b)\), area = \(a \times b\).
  5. Step 5 — Write the units! Area always uses square units (\(\text{cm}^2\), \(\text{m}^2\)) — perimeter uses length units.

Worked Examples

Example 1: Perimeter of a Rectangle

Problem: A rectangle has length \(8\) cm and width \(5\) cm. Calculate its perimeter.

Solution:

  1. \(P = 2 \times (a + b) = 2 \times (8 + 5)\).
  2. \(8 + 5 = 13\).
  3. \(P = 2 \times 13 = 26\) cm.

Answer: \( P = 26 \) cm.

Example 2: Area of a Rectangle

Problem: Same rectangle: length \(8\) cm, width \(5\) cm. Calculate its area.

Solution:

  1. \(S = a \times b = 8 \times 5 = 40\).
  2. Units: cm × cm = square centimetres.

Answer: \( S = 40 \) cm\(^2\).

Example 3: Square — Area and Perimeter

Problem: A square has a side of \(6\) m. Calculate its area and perimeter.

Solution:

  1. Perimeter: \(P = 4 \times 6 = 24\) m.
  2. Area: \(S = 6^2 = 36\) square metres.

Answer: \( P = 24 \) m, \( S = 36 \) m\(^2\).

Example 4: Finding the Missing Side from the Area

Problem: A rectangle has area \(48\) cm\(^2\) and length \(8\) cm. What is its width?

Solution:

  1. \(S = a \times b\), so \(b = S \div a\).
  2. \(b = 48 \div 8 = 6\) cm.
  3. Check: \(8 \times 6 = 48\). Correct!

Answer: The width is \(6\) cm.

Example 5: Word Problem — Tiling a Floor

Problem: A room is \(5\) m long and \(4\) m wide. How many \(1\) m² tiles are needed to cover the floor?

Solution:

  1. Area of the room: \(5 \times 4 = 20\) square metres.
  2. Each tile = \(1\) m\(^2\).
  3. Number of tiles = area ÷ area per tile = \(20 \div 1 = 20\).

Answer: \(20\) tiles are needed.

Common Mistakes

✗ Common mistake: Mixing up area and perimeter: calculating \(a \times b\) when the perimeter is asked for.

✓ The correct way: Perimeter = the border (string around the outside) = \(2(a+b)\). Area = the inside space = \(a \times b\). Ask yourself: "Am I painting the inside (area) or measuring the edge (perimeter)?"

✗ Common mistake: Forgetting to write square units for area: writing "40 cm" instead of "40 cm²".

✓ The correct way: Area is always measured in square units. cm × cm = cm². If the sides are in metres → the area is in m².

✗ Common mistake: Writing \(a + b\) for perimeter instead of \(2(a+b)\) — forgetting the opposite sides.

✓ The correct way: A rectangle has 4 sides: two lengths and two widths. So \(P = a + b + a + b = 2a + 2b = 2(a+b)\).

Practice Tips

  • Tip — Memory trick: Perimeter = "perimeter" = all the way around = add the sides. Area = "area" = the flat inside = multiply.
  • Tip — For a square: all sides are equal! Perimeter = 4 × side. Easy to remember.
  • Tip — Always check units: if the question uses metres, the area answer is in m² and the perimeter answer is in m.
  • Tip — For tiling or painting problems: that's always area. For fencing, framing, or sewing around the edge: that's always perimeter.

Summary and Key Formulas

  • Perimeter of a rectangle: \(P = 2(a+b)\) — length units.
  • Area of a rectangle: \(S = a \times b\) — square units (\(\text{cm}^2, \text{m}^2\)).
  • Square: \(P = 4a\), \(S = a^2\).
  • Finding a missing side: \(b = S \div a\) (if area is known), or \(b = P/2 - a\) (if perimeter is known).