Area and Perimeter Word Problems

Formulas earn their keep when a measurement is missing. If you know the area or the perimeter but not one of the sides, the formula becomes an equation — and solving it recovers the side you need.

Concept The two formulas

Area:

A = l \times w

Perimeter:

P = 2l + 2w

Read left to right they compute a result. Read the other way, with the result known and a side unknown, they become equations to solve.

Concept The situation

Three shops stand side by side on a street, each 2 m wide. Together they form a single rectangle with

3 \times 2 = 6

m of frontage. The depth is unknown.

Example Given the area

The permitted area is 18 m². How deep may the shops be?

Let

x

be the depth. The frontage is 6 m.

6x = 18

Divide both sides by 6:

x = 3

Check:

6 \times 3 = 18

⟹ a depth of 3 m

Example Given the perimeter

Now suppose the permitted perimeter is 20 m instead. How deep may the shops be?

Let

y

be the depth, with the frontage still 6 m.

2 \times 6 + 2y = 20

Simplify:

12 + 2y = 20

Subtract 12 from both sides:

2y = 8

Divide both sides by 2:

y = 4

Check:

2 \times 6 + 2 \times 4 = 12 + 8 = 20

⟹ a depth of 4 m

This one needed two steps rather than one, because the perimeter formula contains both a multiplier and a constant.

Concept Why the answers differ

Same shops, same frontage — yet one constraint permits 3 m of depth and the other permits 4 m. The two limits measure different things.

An area limit caps the floor space inside.

A perimeter limit caps the boundary around.

With a depth of 4 m the area would be $6 \times 4 = 24$ m², well over the 18 m² allowed in the first problem. So always check which quantity the constraint actually restricts before choosing a formula.

Note Mistakes to avoid

Using the area formula when the limit is on the perimeter, or the reverse.

Forgetting to combine the three shop widths into one frontage of 6 m.

Dividing by 2 before subtracting the constant in the perimeter equation.

Doubling only one side in P = 2l + 2w.

Reporting the answer without its unit, or with the wrong one.

Summary

A formula with a missing side becomes an equation to solve.
An area constraint gives a one-step equation: 6x = 18, so x = 3.
A perimeter constraint gives a two-step equation: 12 + 2y = 20, so y = 4.
Clear the constant before dividing by the coefficient.
Area limits floor space; perimeter limits the boundary — they give different answers.