When a shape's side lengths are given as algebraic expressions rather than numbers, you find the perimeter by adding all the sides and collecting like terms. This produces an algebraic expression for the perimeter, which you can simplify and then evaluate by substituting a value for the variable.
What does it mean to write an expression for the perimeter?
The perimeter of any shape is the total length of its boundary — the sum of all its side lengths. When sides are labelled with expressions like (2x + 3) or (x − 1), you add those expressions together and simplify by collecting like terms.
The result is a single algebraic expression that gives the perimeter for any value of x. You have not solved an equation — you have formed a formula.
How do you write and simplify a perimeter expression for a rectangle?
Worked example 1: A rectangle has length (3x + 5) cm and width (x + 2) cm.
- Add all four sides: (3x + 5) + (x + 2) + (3x + 5) + (x + 2)
- Collect x terms: 3x + x + 3x + x = 8x
- Collect constants: 5 + 2 + 5 + 2 = 14
- Perimeter = (8x + 14) cm
Shortcut: for a rectangle, Perimeter = 2 × (length + width) = 2 × [(3x + 5) + (x + 2)] = 2 × (4x + 7) = 8x + 14 ✓
How do you find the perimeter of a triangle with algebraic sides?
Worked example 2: A triangle has sides (2x − 1) cm, (x + 4) cm and (3x + 2) cm.
| Step | Working |
|---|---|
| Write the sum | (2x − 1) + (x + 4) + (3x + 2) |
| Collect x terms | 2x + x + 3x = 6x |
| Collect constants | −1 + 4 + 2 = 5 |
| Perimeter | (6x + 5) cm |
Check: if x = 3, sides are 5, 7 and 11 cm. Perimeter = 23 cm. Expression: 6(3) + 5 = 18 + 5 = 23 ✓
How do you use the perimeter to form an equation?
Sometimes the perimeter is given as a number or expression, and you need to find x.
Worked example 3: The rectangle from Worked example 1 has a perimeter of 38 cm. Find x and then find the actual dimensions.
- Expression for perimeter: 8x + 14
- Set equal to 38: 8x + 14 = 38
- Subtract 14: 8x = 24
- Divide by 8: x = 3
- Length = 3(3) + 5 = 14 cm; Width = 3 + 2 = 5 cm
This combines three skills: writing the expression, simplifying it, and solving the resulting equation.
How do you find an expression for the area of a rectangle?
For area, multiply the two expressions (expand the brackets rather than just adding).
Worked example 4: Area of the rectangle with length (3x + 5) cm and width (x + 2) cm.
Area = (3x + 5)(x + 2)
Expand using FOIL or the grid method:
- 3x × x = 3x²
- 3x × 2 = 6x
- 5 × x = 5x
- 5 × 2 = 10
Area = 3x² + 6x + 5x + 10 = (3x² + 11x + 10) cm²
Note that the area expression contains x² because it is a two-dimensional measurement.
What mistakes should you avoid?
- Forgetting to include all sides. A rectangle has four sides; a triangle has three. Draw the shape and label each side before adding.
- Collecting unlike terms. x terms and constant terms must be kept separate. 3x + 5 cannot be simplified further.
- Confusing perimeter (add all sides) with area (multiply dimensions). Perimeter involves addition, giving a linear expression; area involves multiplication, often giving a quadratic.
- Not checking with a value of x. Always substitute a simple value (e.g. x = 1 or x = 2) into both the original sides and the simplified expression to verify they give the same perimeter.
Frequently asked questions
Can the perimeter expression have a negative coefficient?
Yes, if one of the sides contains a subtraction. For example, if a side is (2x − 3), collecting the constant gives −3, which may reduce the total constant in the perimeter. Side lengths must still be positive numbers, so when substituting x, check that each individual side length is positive.
What if the shape has more than four sides?
The method is identical — add every side, then collect like terms. For a regular hexagon (six equal sides) with side length (2x + 1), the perimeter is 6 × (2x + 1) = 12x + 6. For an irregular polygon, list every side and add them systematically, sorting into x terms and constants.
My perimeter expression for a square gives (4x + 12). What does this tell me?
It tells you that the perimeter depends on x. The 4x part changes with x; the 12 is a fixed contribution. If x = 5, the perimeter is 4(5) + 12 = 32 units. It also means each side of the square is (x + 3) units (divide the whole perimeter by 4: (4x + 12)/4 = x + 3). This is a useful reverse check.
Do I need to write the unit in the expression?
Yes, always include the unit on the final answer. If the sides are given in centimetres, the perimeter is in centimetres. Writing "(8x + 14)" without "cm" is an incomplete answer. Area expressions need cm² (or m², etc.) as the unit.
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