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Constructing expressions from a geometric model — 0580

Define a variable and translate a description before calculating.

Core & Extended · Syllabus C2.1, C2.5, E2.1, E2.5

Notes and key methods

Choose a variable with units and write each stated relationship in symbols. A rectangle with width x and length ax+b has area x(ax+b), while its perimeter is twice the sum of its dimensions.

Lengths, areas and perimeters have different units and formulas. Brackets retain the grouping described in words. Check the model with a simple permitted value, and keep every length positive when solving an equation obtained from it.

Check the conditions

Three worked examples

Example 1 · Constructing expressions from a geometric model

A rectangle has width x cm and a length that is 7 cm more than 4 times its width. Assume x>0.

  1. Construct and simplify an expression for the area.
  2. Construct and simplify an expression for the perimeter.
  3. The perimeter is \(54\) cm. Find x and both dimensions.
  4. Write the perimeter as Cx+D. Find D.
Show the reasoning

(a)

Multiply the width x by the length \(4 x + 7\). Expanding gives \(4 x^{2} + 7 x\).

Answer: \(A=4 x^{2} + 7 x\) \(\displaystyle \mathrm{cm}^{2}\).

(b)

Add one length and one width, then double the whole sum.

Answer: \(P=10 x + 14\) cm.

(c)

Set \(10 x + 14\) equal to \(54\). Solving gives \(\displaystyle x=4\), and both resulting lengths are positive.

Answer: \(x=4\), width 4 cm, length 23 cm.

(d)

Translate the described dimensions, then use either the area product or twice the sum of the sides.

\(14\)

Answer: \(14\).

Example 2 · Constructing expressions from a geometric model

A rectangle has width x cm and a length that is 8 cm more than 5 times its width. Assume x>0.

  1. Construct and simplify an expression for the area.
  2. Construct and simplify an expression for the perimeter.
  3. The perimeter is \(76\) cm. Find x and both dimensions.
  4. Write the perimeter as Cx+D. Find D.
Show the reasoning

(a)

Multiply the width x by the length \(5 x + 8\). Expanding gives \(5 x^{2} + 8 x\).

Answer: \(A=5 x^{2} + 8 x\) \(\displaystyle \mathrm{cm}^{2}\).

(b)

Add one length and one width, then double the whole sum.

Answer: \(P=12 x + 16\) cm.

(c)

Set \(12 x + 16\) equal to \(76\). Solving gives \(\displaystyle x=5\), and both resulting lengths are positive.

Answer: \(x=5\), width 5 cm, length 33 cm.

(d)

Translate the described dimensions, then use either the area product or twice the sum of the sides.

\(16\)

Answer: \(16\).

Example 3 · Constructing expressions from a geometric model

A rectangle has width x cm and a length that is 9 cm more than 6 times its width. Assume x>0.

  1. Construct and simplify an expression for the area.
  2. Construct and simplify an expression for the perimeter.
  3. The perimeter is \(102\) cm. Find x and both dimensions.
  4. Write the perimeter as Cx+D. Find D.
Show the reasoning

(a)

Multiply the width x by the length \(6 x + 9\). Expanding gives \(6 x^{2} + 9 x\).

Answer: \(A=6 x^{2} + 9 x\) \(\displaystyle \mathrm{cm}^{2}\).

(b)

Add one length and one width, then double the whole sum.

Answer: \(P=14 x + 18\) cm.

(c)

Set \(14 x + 18\) equal to \(102\). Solving gives \(\displaystyle x=6\), and both resulting lengths are positive.

Answer: \(x=6\), width 6 cm, length 45 cm.

(d)

Translate the described dimensions, then use either the area product or twice the sum of the sides.

\(18\)

Answer: \(18\).

Common mistakes

Practise with answer checking

Original exam-style problems

Write complete solutions before opening the worked solution. The marking guidance rewards reasoning as well as answers.

Examination practice · 2020–2026

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