Understanding Perimeter
Perimeter is the total length around the boundary of a two-dimensional shape. To find the perimeter, you add together the lengths of all the sides.
For example, a rectangle has four sides: two lengths and two widths. The perimeter is the sum of these four sides.
The perimeter is always expressed in units of length, such as centimetres (cm), metres (m), or millimetres (mm).
- Perimeter = sum of all side lengths
- Units of perimeter are linear (e.g., cm, m)
Calculating Perimeter of Common Shapes
Each shape has a specific method to calculate perimeter based on its sides.
Rectangle: Since opposite sides are equal, perimeter = 2 × (length + width).
Triangle: Add the lengths of all three sides. No formula shortcut applies unless it is an equilateral triangle (all sides equal). Example: sides 5 cm, 7 cm, and 8 cm give perimeter = 5 + 7 + 8 = 20 cm. Parallelogram: Opposite sides are equal, so perimeter = 2 × (base + side). Trapezium: Sum all four sides as they can be different lengths.
- Rectangle perimeter = 2(length + width)
- Triangle perimeter = sum of all sides
- Parallelogram perimeter = 2(base + side)
- Trapezium perimeter = sum of all sides
Understanding Area
Area measures the size of the surface enclosed within a shape. It is expressed in square units such as cm², m², or mm².
Area formulas depend on the shape because each shape covers space differently.
Calculating area involves multiplying lengths in a way that accounts for the shape’s geometry, often involving base and height measurements.
- Area measures surface inside a shape
- Units are squared (e.g., cm², m²)
Calculating Area of Rectangles, Parallelograms, and Trapeziums
Rectangle: Area = length × width. This works because the rectangle is a grid of unit squares equal to length times width.
Parallelogram: Area = base × height. The height is the perpendicular distance from the base to the opposite side, not the length of the slanting side.
Trapezium: Area = ½ × (sum of parallel sides) × height. The height is the perpendicular distance between the two parallel sides (called bases).
- Rectangle area = length × width
- Parallelogram area = base × height
- Trapezium area = ½ × (base1 + base2) × height
Worked Examples
Example 1: Rectangle perimeter and area Length = 8 cm, Width = 5 cm Perimeter = 2(8 + 5) = 2 × 13 = 26 cm Area = 8 × 5 = 40 cm² Example 2: Triangle perimeter Sides = 6 cm, 7 cm, 9 cm Perimeter = 6 + 7 + 9 = 22 cm Example 3: Parallelogram area Base = 10 cm, Height = 6 cm Area = 10 × 6 = 60 cm² Example 4: Trapezium area Parallel sides = 7 cm and 5 cm, Height = 4 cm Area = ½ × (7 + 5) × 4 = ½ × 12 × 4 = 24 cm²
- Show substitution clearly
- Include units in answers
- Check calculations for accuracy
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Calculate perimeter | Add all side lengths or use formula for regular shapes | Rectangle with length 8 cm and width 5 cm: Perimeter = 2(8 + 5) = 26 cm | Write formula, substitute values, calculate, include units |
| Calculate area | Use shape-specific formula involving length, width, base, height | Trapezium with bases 7 cm and 5 cm, height 4 cm: Area = ½(7 + 5) × 4 = 24 cm² | Identify correct formula, substitute carefully, show working, include units |
| Identify height | Height is perpendicular distance to base, not side length | Parallelogram with slant side 8 cm but height 6 cm: use height 6 cm in area formula | Measure or identify perpendicular height accurately to avoid errors |
| Sum sides for perimeter | For irregular shapes like trapezium, add all sides individually | Trapezium sides 3 cm, 5 cm, 7 cm, 4 cm: Perimeter = 3 + 5 + 7 + 4 = 19 cm | Do not confuse perimeter with area; sum all sides for perimeter |
Active recall
Quick check
1. What is the perimeter of a rectangle with length 12 cm and width 7 cm?
Perimeter = 2 × (12 + 7) = 2 × 19 = 38 cm.
2. Calculate the area of a parallelogram with base 9 cm and height 5 cm.
Area = base × height = 9 × 5 = 45 cm².
3. A trapezium has parallel sides 6 cm and 10 cm, and height 4 cm. What is its area?
Area = ½ × (6 + 10) × 4 = ½ × 16 × 4 = 32 cm².
4. Find the perimeter of a triangle with sides 8 cm, 6 cm, and 5 cm.
Perimeter = 8 + 6 + 5 = 19 cm.