Perimeter and Area of Plane Figures
Learn to measure the distance around a shape and the space inside it. Revise unit conversions, rectangles, triangles, circles, compound figures, parallelograms and trapeziums.
Step-by-step explanations, clear diagrams, short self-checks and two free printable practice PDFs with worked solutions. No sign-up required.
What you will learn in Chapter 12
Convert units
Change between cm, m, km and their square units correctly.
Find perimeter and area
Choose the correct formula for rectangles, triangles and circles.
Solve compound shapes
Combine or subtract areas and trace the actual outside boundary.
Use perpendicular height
Calculate areas of parallelograms and trapeziums, including missing lengths.
Jump to a lesson
Length and area units are not converted the same way
Perimeter is a length, so its unit is cm, m or km. Area measures a surface, so its unit is cm², m² or km². When changing area units, square the length conversion factor.
Length
1 m = 100 cm
1 km = 1,000 m
Example: 4.35 m = 4.35 × 100 = 435 cm.
Area
1 m² = 10,000 cm²
1 cm² = 100 mm²
1 km² = 1,000,000 m²
Example: 0.072 m² = 0.072 × 10,000 = 720 cm².
Perimeter goes around. Area fills the inside.
To find perimeter, add lengths along the boundary. To find area, measure how much surface a shape covers. These are different quantities and need different units.
Rectangle
P = 2(l + w) A = lw
Perimeter = 2(10 + 6) = 32 cm
Area = 10 × 6 = 60 cm²
Square
P = 4s A = s²
Perimeter = 4 × 8 = 32 cm
Area = 8² = 64 cm²
| Shape | Perimeter / circumference | Area |
|---|---|---|
| Rectangle | 2(l + w) | lw |
| Square | 4s | s² |
| Triangle | Sum of three side lengths | ½ × base × perpendicular height |
| Circle | 2πr = πd | πr² |
| Parallelogram | 2(a + b), where a and b are adjacent sides | base × perpendicular height |
| Trapezium | Sum of all four side lengths | ½ × (sum of parallel sides) × perpendicular height |
Use the perpendicular height—not just any sloping side
Area = ½ × base × height
The height must be at 90° to the chosen base. The perimeter is found separately by adding the three sides.
Example: base = 12 cm and perpendicular height = 7 cm.
Area = ½ × 12 × 7 = 42 cm²
For a separate triangle with sides of 10 cm, 11 cm and 12 cm, its perimeter is 10 + 11 + 12 = 33 cm.
Worked example: find a missing height
A triangle has area 63 cm² and base 18 cm. Find its perpendicular height.
63 = ½ × 18 × h = 9h, so h = 7 cm.
Radius, diameter, circumference and area
Circle
C = 2πr = πd
A = πr²
For r = 7 cm and π = 3.142:
C = 2 × 3.142 × 7 = 43.988 ≈ 44.0 cm (3 s.f.)
A = 3.142 × 7² = 153.958 ≈ 154 cm² (3 s.f.)
Semicircle
A = ½πr²
P = πr + 2r
For r = 4 cm:
Area = ½ × 3.142 × 4² = 25.136 cm²
Complete perimeter = 3.142 × 4 + 8 = 20.568 cm.
Add parts, subtract gaps and trace the outside edge
Compound shapes combine simple figures. For area, divide the shape into manageable pieces or subtract an unwanted region. For perimeter, measure only the actual outside boundary.
Sketch and label
Mark all given lengths. Work out any missing dimensions.
Choose the area method
Add non-overlapping areas or subtract an opening from a larger figure.
Trace the perimeter
Count outer straight and curved edges only; ignore internal joins.
Example A · Corner removed
Area = 10 × 8 − 4 × 3 = 68 cm².
The two removed outer lengths are replaced by two equal inner lengths, so perimeter remains 36 cm.
Example B · Path outside a lawn
Outer area = 17 × 9 = 153 m².
Inner area = 13 × 5 = 65 m².
Path area = 153 − 65 = 88 m².
A parallelogram uses a perpendicular height
Area = base × perpendicular height
Perimeter = 2(a + b)
For a base of 10 cm and perpendicular height of 6 cm:
Area = 10 × 6 = 60 cm²
The sloping side is not the height. If the same parallelogram has area 84 cm² and you choose a 12 cm side as the base, its corresponding height is 84 ÷ 12 = 7 cm.
Add the parallel sides before multiplying by height
Area = ½ × (a + b) × h
Here, a and b are the lengths of the two parallel sides, while h is the perpendicular distance between them.
Example: parallel sides = 8 cm and 14 cm; height = 5 cm.
A = ½ × (8 + 14) × 5 = 55 cm²
Worked example: missing parallel side
Area = 96 cm², height = 8 cm, one parallel side = 10 cm. Let the other side be x.
96 = ½ × (10 + x) × 8
96 = 4(10 + x)
24 = 10 + x
x = 14 cm
Apply the ideas to multi-step problems
Example 1 · Find dimensions from perimeter
A rectangle has perimeter 58 m. Its length is 7 m longer than its width. Find its area.
Reveal full working
Let width = x m and length = (x + 7) m.
2[x + (x + 7)] = 58
4x + 14 = 58
x = 11
Width = 11 m, length = 18 m.
Area = 11 × 18 = 198 m².
Example 2 · Area outside a circle
A circle of radius 5 cm lies inside a square of side 12 cm. Find the area outside the circle but inside the square, to 3 significant figures.
Reveal full working
Square = 12² = 144 cm².
Circle = 3.142 × 5² = 78.55 cm².
Required area = 144 − 78.55 = 65.45 cm².
65.5 cm² (3 s.f.).
Example 3 · Missing parallelogram height
A parallelogram has area 154 cm² and base 14 cm. Find its perpendicular height.
Reveal full working
Area = base × height.
154 = 14 × h
h = 154 ÷ 14 = 11 cm.
Example 4 · Shaded circular region
A square garden has side 14 m and a circular flowerbed of radius 5 m. Find the uncovered area to 3 significant figures.
Reveal full working
Square area = 14² = 196 m².
Circle area = 3.142 × 5² = 78.55 m².
Uncovered area = 196 − 78.55 = 117.45 m² ≈ 117 m² (3 s.f.).
Five mistakes to avoid
1. Mixing length and area units
Perimeter: cm or m. Area: cm² or m². Convert measurements before applying formulas.
2. Using a sloping height
Triangles, parallelograms and trapeziums require a height at 90° to the chosen base.
3. Forgetting the diameter
Complete perimeter of a semicircle is πr + 2r, not just πr.
4. Counting hidden/internal edges
A shared joining edge is not part of the outside perimeter of a combined shape.
5. Rounding too early
Keep accurate intermediate values, then round the final answer as required.
Before submitting
Show formula → substitution → calculation → correct unit. Check that your answer is reasonable.
Can you answer these six questions?
Choose one answer for each question, then select Check answers. Explanations appear after checking. Nothing is sent or stored.
Try these before opening the worksheets
Write down your working, then reveal the answers to check your reasoning.
Question A · Unit conversion
Convert 4,200 cm² to m².
Show answer
Question B · Semicircle
A semicircle has radius 3 cm. Find its complete perimeter, using π = 3.142.
Show answer
Question C · Compound area
Remove a 3 cm × 2 cm rectangle from a 12 cm × 7 cm rectangle. Find the remaining area.
Show answer
Question D · Trapezium
A trapezium has area 84 cm², height 7 cm and one parallel side 10 cm. Find the other parallel side.
Show answer
Download Chapter 12 worksheets with worked solutions
Use the learning workbook to revise each topic first, then attempt the supplementary practice for more challenging, multi-step questions. Both resources are free to open, save and print.
Perimeter and Area — Chapter Workbook
Concept explanations, formulas, labelled diagrams, worked examples, topic-by-topic practice, mixed challenge questions and solutions.
Recommended first: learn → practise → review answers.
Open Free Workbook PDF ↗Additional Questions with Full Worked Solutions
Further questions on unit conversions, circles, shaded regions, paths, parallelograms, trapeziums and challenging compound figures.
Recommended next: attempt independently before reading the solutions.
Open Supplementary PDF ↗A simple Chapter 12 learning sequence
- Revise units first: distinguish length conversions from area conversions.
- Learn the basic formulas: rectangle, square, triangle and circle.
- Practise reasoning about boundaries: trace the outside of compound figures and identify excluded regions.
- Focus on perpendicular height: apply it to triangles, parallelograms and trapeziums.
- Work through PDF 1: complete topic practice before checking worked solutions.
- Use PDF 2 for extension: attempt its multi-step and shaded-region questions.
Ready to practise perimeter and area?
Download the free Chapter 12 worksheets, show each step clearly and revisit any formula or diagram that needs more practice.
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