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 (\(a = b\)):
\[ P_{\text{square}} = 4a \qquad S_{\text{square}} = a^2 \]
| Measure | Formula — Rectangle | Formula — Square |
|---|---|---|
| Perimeter | \(2(a+b)\) | \(4a\) |
| Area | \(a \times b\) | \(a^2\) |
Solution Steps
- Step 1 — Identify the shape: rectangle (length ≠ width in general) or square (all sides equal).
- Step 2 — Read what is being asked: perimeter (the boundary line) or area (the inside surface).
- Step 3 — Check the units: are all measurements in the same unit? If not, convert first (e.g., cm to m).
- Step 4 — Calculate using the formula: perimeter = \(2(a+b)\), area = \(a \times b\).
- 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:
- \(P = 2 \times (a + b) = 2 \times (8 + 5)\).
- \(8 + 5 = 13\).
- \(P = 2 \times 13 = 26\) cm.
Answer: \( P = 26 \) cm.
Example 2: Area of a Rectangle
Problem: Same rectangle: length \(8\) cm, width \(5\) cm. Calculate its area.
Solution:
- \(S = a \times b = 8 \times 5 = 40\).
- 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:
- Perimeter: \(P = 4 \times 6 = 24\) m.
- 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:
- \(S = a \times b\), so \(b = S \div a\).
- \(b = 48 \div 8 = 6\) cm.
- 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:
- Area of the room: \(5 \times 4 = 20\) square metres.
- Each tile = \(1\) m\(^2\).
- 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).