Area and Perimeter: Formulas, Examples & Worksheets

Start with the meaning

Know how to add lengths, multiply whole numbers and recognise a rectangle. This guide uses rectangles and squares; it does not teach circle area or the area of arbitrary polygons.

A 6 cm by 4 cm rectangle contains 24 unit squares. Its boundary measures 6 + 4 + 6 + 4 = 20 cm.

Trace the boundary for perimeter

Imagine a thin string going once around the outside edge. For a rectangle of length six centimetres and width four centimetres, add 6 + 4 + 6 + 4 = 20 cm.

Opposite sides of a rectangle are equal, so P = 2 × (length + width). This formula is a short way to count all four sides, not a rule for every shape.

Cover the inside for area

Cover the same rectangle with 1 cm by 1 cm squares. Six squares fit along each row and there are four rows, giving 6 × 4 = 24 square centimetres.

Area uses square units because each counted tile has both length and width. Multiplying two lengths in centimetres produces cm². Perimeter remains a length in cm.

Equal perimeter does not guarantee equal area

A 6 by 4 rectangle has perimeter 20 and area 24. An 8 by 2 rectangle also has perimeter 20, but its area is 16. Use the same length units for both.

This counterexample is enough to disprove the claim that equal perimeter always means equal area. Drawing both on squared paper makes the difference visible.

Read the context and match units

Fencing usually follows a boundary; floor covering occupies a surface. Draw what the material must cover rather than deciding only from a keyword. A border strip with width, for example, may itself have an area.

Convert dimensions to the same unit before using a formula. A 2 m by 50 cm rectangle can be treated as 2 m by 0.5 m, giving area 1 m² and perimeter 5 m.

See it, then explain it

Two illustrated examples, explained step by step

Same fence, different areas: which park gives more space?

Swipe across the diagram to see each part.

A 5 m by 1 m park and a 4 m by 2 m park both use 12 m of fencing. Their usable areas are different: 5 m² and 8 m².

Trace park 1: 5 + 1 + 5 + 1 = 12 m. Its area is 5 × 1 = 5 m².
Trace park 2: 4 + 2 + 4 + 2 = 12 m. Its area is 4 × 2 = 8 m².
Compare the spaces: 8 − 5 = 3 m² more in park 2, with the same boundary length.

Perimeter measures the boundary; area measures the surface enclosed. Equal boundary lengths do not force equal surfaces. The unit squares in the diagrams have the same scale, so you can compare the covered space directly.

Save this illustration on Pinterest · Download the illustrated summary

Grass or fencing? Measure the part you need

Swipe across the diagram to see each part.

For one 5 m by 3 m garden, grass covers the inside and fencing follows the outside. Those are two measurements of the same shape.

Count five unit squares in each of three rows: 5 × 3 = 15 m² of grass.
Add the four sides: 5 + 3 + 5 + 3 = 16 m of fencing.
Label the answers: square metres for surface area, metres for boundary length.

Draw what the material covers before choosing a calculation. If the fence has a 1 m gate, subtract that gap from the 16 m boundary to buy 15 m of fencing; the grass area stays 15 m². The equal numbers here still describe different quantities.

Save this illustration on Pinterest · Download the illustrated summary

Worked examples, step by step

Example 1: A rectangle is 9 cm long and 3 cm wide. Find perimeter and area.

Perimeter counts all edges: 9 + 3 + 9 + 3 = 24 cm.
Area counts unit squares: 9 × 3 = 27 cm².
Use length units for the boundary and square units for the surface.

Example 2: A rectangle has area 35 cm² and width 5 cm. Find its length.

Area = length × width, so length × 5 = 35.
Length = 35 ÷ 5 = 7 cm.
Check 7 × 5 = 35 cm². The answer is a length, not an area.

Example 3: A square has perimeter 28 m. Find its area.

Four equal sides total 28 m, so each side is 28 ÷ 4 = 7 m.
Area is 7 × 7 = 49 m².
Do not square the perimeter. Find the side length first.

Choose a formula, match the units, and handle a composite shape

Area and perimeter answer different questions. Before calculating, mark the surface you want to cover or trace the boundary you want to measure. For the formulas below, l means rectangle length, w means rectangle width and s means square side length. All lengths in one calculation must use the same unit.

Choose a method that matches the question
Shape or task	Method	What the answer measures
Rectangle area	A = l × w	Covered surface, in square units
Rectangle perimeter	P = 2 × (l + w)	Total boundary length
Square area and perimeter	A = s × s; P = 4 × s	Surface and boundary respectively
Missing rectangle length	l = A ÷ w	A length, not an area

Changing a length unit changes two dimensions in an area. Since 1 m = 100 cm, a 1 m by 1 m square contains 100 × 100 = 10,000 square centimetres. Therefore 1 m² = 10,000 cm², not 100 cm². Convert side lengths first when that makes the calculation easier to follow.

Worked example: An 8 m by 5 m rectangular garden has a 3 m by 2 m corner removed. Find the remaining area.

Remove only the labelled corner. The remaining shape can also be split into two non-overlapping rectangles.

Start with the whole rectangle: 8 × 5 = 40 m².
The removed corner covers 3 × 2 = 6 m². Subtract it once: 40 − 6 = 34 m².
Check by splitting the remaining L-shape into a 5 m by 5 m rectangle and a 3 m by 3 m rectangle: 25 + 9 = 34 m².

For this corner cut-out, the perimeter remains 26 m: the removed 3 m and 2 m outside edges are replaced by a 3 m and a 2 m inside edge. Do not extend that result to every cut-out. A notch cut into the middle of one side can increase the perimeter; trace the actual boundary.

Try a fresh example, then explain your check

A rectangle is 250 cm long and 2 m wide. Find its area in m² and perimeter in m.
Show answer and reasoning
250 cm = 2.5 m. Area = 2.5 × 2 = 5 m²; perimeter = 2 × (2.5 + 2) = 9 m.
A square has area 49 cm². Find its perimeter.
Show answer and reasoning
A side is 7 cm because 7 × 7 = 49. Its perimeter is 4 × 7 = 28 cm. Do not multiply the area by four.
A 6 m by 4 m lawn needs edging except for a 1.5 m opening. How much edging is needed?
Show answer and reasoning
The complete boundary is 2 × (6 + 4) = 20 m. The opening needs no edging, so buy 20 − 1.5 = 18.5 m, before any allowance for joins or waste.

Questions about choosing and checking a method

Can two shapes have the same area but different perimeters?

Yes. A 6 cm by 2 cm rectangle and a 4 cm by 3 cm rectangle each cover 12 cm². Their perimeters are 16 cm and 14 cm. Equal area does not determine a unique boundary length.

When is subtracting a smaller rectangle a valid area method?

It works when the larger rectangle is exactly the required shape plus the removed rectangle, with no gaps or overlaps. Label the missing rectangle dimensions from the drawing; do not estimate them from how the picture looks.

Common mistakes and useful repairs

Multiplying to find every perimeter

Multiplication can shorten a sum of equal sides, but perimeter is still the total boundary length. For a rectangle, length × width finds area.

Assuming a larger area always means a larger perimeter

A 4 by 4 square has area 16 and perimeter 16. A 1 by 10 rectangle has smaller area 10 but larger perimeter 22, using matching units.

Try before revealing

Quick understanding check

Read the check answers

cm². Area measures a two-dimensional surface in square units.

16 cm. 5 + 3 + 5 + 3 = 16 cm.

Area and perimeter: the difference, examples and practice — practice sheet

Work on paper before opening an answer. Print this question sheet with or without a separate worked key; no account is needed.

Include worked answer key

Name: ____________________ Date: __________

Find the perimeter of an 8 cm by 5 cm rectangle.
Show answer and reasoning
26 cm. 2 × (8 + 5) = 26.
Find its area.
Show answer and reasoning
40 cm². 8 × 5 = 40 unit squares.
A square has side 6 m. Find perimeter and area.
Show answer and reasoning
24 m and 36 m². Four six-metre sides give 24 m; six rows of six squares give 36 m².
A rectangle has area 48 cm² and width 6 cm. Find its length.
Show answer and reasoning
8 cm. 48 ÷ 6 = 8.
A rectangle has perimeter 30 cm and length 9 cm. Find its width.
Show answer and reasoning
6 cm. The two lengths use 18 cm, leaving 12 cm for two equal widths.
Find the area of a rectangle 3 m long and 40 cm wide.
Show answer and reasoning
1.2 m². Convert 40 cm to 0.4 m, then 3 × 0.4 = 1.2.
Give two rectangles with perimeter 16 cm but different areas.
Show answer and reasoning
For example, 5 by 3 and 6 by 2. Both perimeters are 16 cm. Areas are 15 cm² and 12 cm².
A 7 m by 4 m garden needs fencing with a 1 m gap for a gate. How much fence?
Show answer and reasoning
21 m. Full perimeter is 22 m. Subtract the one-metre opening; the gate is not part of the fence length requested.

Area and perimeter: the difference, examples and practice — worked answer key

26 cm. 2 × (8 + 5) = 26.
40 cm². 8 × 5 = 40 unit squares.
24 m and 36 m². Four six-metre sides give 24 m; six rows of six squares give 36 m².
8 cm. 48 ÷ 6 = 8.
6 cm. The two lengths use 18 cm, leaving 12 cm for two equal widths.
1.2 m². Convert 40 cm to 0.4 m, then 3 × 0.4 = 1.2.
For example, 5 by 3 and 6 by 2. Both perimeters are 16 cm. Areas are 15 cm² and 12 cm².
21 m. Full perimeter is 22 m. Subtract the one-metre opening; the gate is not part of the fence length requested.

Keep practising · suggested Grade 5

Worksheets and revision: Area & perimeter

Practise measuring boundaries, covering surfaces, matching units and comparing shapes.

Two worksheet pages, 12 questions and separate worked answers. Try the questions before checking. Downloads are PNG images; the ZIP contains the complete image pack.

Worked answer sheets: Answers 1

Open the 12 additional questions and worked solutions

Open each question after you have attempted it. Refer to the matching worksheet diagram where indicated.

1. Count the squares

Find the area and perimeter of a grid four squares wide and three squares tall. Each square is 1 cm by 1 cm.