Algebraic Area and Perimeter GCSE: Forming and Solving Equations

How do you form a perimeter equation?

A perimeter equation adds the algebraic expressions for all sides of the shape.

Worked example 1: A rectangle has length (3x + 2) cm and width (x + 5) cm. Its perimeter is 46 cm. Find x.

• Perimeter = 2(length + width) = 2(3x + 2 + x + 5) = 2(4x + 7) = 8x + 14
• Set equal to 46: 8x + 14 = 46
• 8x = 32, so x = 4

Check: length = 14 cm, width = 9 cm; perimeter = 2(14 + 9) = 46 cm ✓

Worked example 2: An equilateral triangle has sides (2x − 1) cm, (x + 4) cm, and (3x − 5) cm. Find x.

An equilateral triangle has all sides equal, so:

2x − 1 = x + 4 → x = 5

Check the third side: 3(5) − 5 = 10. And x + 4 = 9... wait. Let me re-read.

Actually: for a general triangle with those side labels, the perimeter equals their sum. If the perimeter is given:

• Perimeter = (2x − 1) + (x + 4) + (3x − 5) = 6x − 2
• If perimeter = 34: 6x − 2 = 34 → x = 6
• Sides: 11, 10, 13 cm — all positive ✓

How do you form an area equation?

For a rectangle, area = length × width. Set the expression equal to the given area and solve — this often produces a quadratic equation.

Worked example 3: A rectangle has length (x + 7) cm and width (x − 2) cm. Its area is 30 cm². Find x.

• Area = (x + 7)(x − 2) = 30
• Expand: x² − 2x + 7x − 14 = 30
• x² + 5x − 14 = 30
• x² + 5x − 44 = 0
• Factorise: (x + 11)(x − 4) = 0
• x = −11 or x = 4

Since x = −11 gives a negative width (−11 − 2 = −13), reject x = −11.

So x = 4: length = 11 cm, width = 2 cm; area = 22 cm² — wait, that gives 22, not 30.

Let me redo: (4 + 7)(4 − 2) = 11 × 2 = 22 ≠ 30.

The quadratic was x² + 5x − 44 = 0. Discriminant = 25 + 176 = 201 — not a perfect square, so the quadratic does not factorise neatly. Let me use a cleaner example.

Corrected Worked example 3: A rectangle has length (x + 6) cm and width x cm. Its area is 40 cm². Find x.

• x(x + 6) = 40
• x² + 6x − 40 = 0
• Factorise: (x + 10)(x − 4) = 0
• x = −10 (rejected, negative length) or x = 4
• Width = 4 cm; length = 10 cm; area = 40 cm² ✓

Key step: always reject negative values that would give a negative side length.

How do you handle compound shapes?

Break the compound shape into rectangles or triangles, write an expression for the area of each part, add them, and set equal to the given total.

Worked example 4: An L-shaped room has two rectangular sections. One section measures (2x) m by 3 m; the other measures (x + 1) m by 2 m. The total floor area is 54 m². Find x.

• Total area = 6x + 2(x + 1) = 6x + 2x + 2 = 8x + 2
• 8x + 2 = 54 → 8x = 52 → x = 6.5
• Check: 6(6.5) = 39; 2(7.5) = 15; total = 54 m² ✓

What if the equation gives a non-integer solution?

Leave x as a fraction or decimal — the GCSE specification allows non-integer answers. However, check that all side lengths are positive before writing your final answer.

What about finding an angle or unknown when area involves trigonometry?

Some GCSE questions combine area with the sine rule for triangles: Area = ½ab sin C. If the area is given and one side is expressed algebraically, substitute and solve. This is Higher tier only.

Frequently asked questions

Why do we reject negative solutions?

A side length must be positive — a negative length has no geometric meaning. When a quadratic gives two solutions, substitute each back to check whether the resulting side lengths are positive. If one solution is negative or leads to a negative side, reject it and state clearly in your working why you have done so.

Can the answer to x be a decimal?

Yes. For example, if the perimeter equation gives 8x = 20, then x = 2.5 is a perfectly valid answer. Always check that the resulting side lengths are positive, even when x is a decimal.

What if I expand the brackets incorrectly when forming the area equation?

Errors in expansion lead to a wrong quadratic and wrong solutions. Check your expansion by substituting a simple value of x (e.g. x = 1) into both the unexpanded and expanded forms — they should give the same result. This spot-check takes seconds and catches sign errors.

How do I know whether to form a linear or quadratic equation?

A perimeter question almost always gives a linear equation (you add expressions, not multiply them). An area question for a rectangle or triangle usually gives a quadratic (area = length × width or ½ base × height involves a product of two algebraic expressions). Look at what operation defines the formula.

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