What you'll learn
This topic covers the calculation of perimeter and area for triangles, quadrilaterals, circles, sectors, and compound shapes—a fundamental skill tested across both Core and Extended CIE IGCSE Mathematics papers. You'll apply formulae, work with exact values involving π, and solve multi-step problems requiring shape decomposition. Questions on this topic typically appear in Paper 2 (calculator) and Paper 4 (Extended), often combined with similarity, trigonometry, or problem-solving contexts.
Key terms and definitions
Perimeter — the total distance around the outside edge of a two-dimensional shape, measured in linear units (cm, m, etc.)
Area — the amount of two-dimensional space enclosed within a shape's boundaries, measured in square units (cm², m², etc.)
Sector — a region of a circle bounded by two radii and an arc, resembling a 'slice' of the circle
Arc length — the distance along the curved part of a circle or sector, calculated as a fraction of the circumference
Compound shape — a two-dimensional figure formed by combining two or more simple shapes (rectangles, triangles, circles, etc.)
Exact form — an answer expressed using π, surds, or fractions rather than decimal approximations
Trapezium — a quadrilateral with exactly one pair of parallel sides (called bases)
Base and perpendicular height — the horizontal side and the vertical distance from the base to the opposite side or vertex, crucial for calculating areas of triangles and quadrilaterals
Core concepts
Triangles
All triangles share the fundamental area formula:
Area = ½ × base × perpendicular height
The perpendicular height must form a 90° angle with the base. In obtuse triangles, you may need to extend the base line to find where the perpendicular meets it.
For the perimeter of any triangle, simply add all three side lengths: a + b + c.
When working with right-angled triangles, Pythagoras' theorem (a² + b² = c²) frequently appears in exam questions to find missing sides before calculating perimeter or area.
Special cases tested at IGCSE level:
- Equilateral triangles: All sides equal, all angles 60°. Area = (√3/4) × side² (rarely required, but useful for Extended tier)
- Isosceles triangles: Two equal sides; perpendicular height often requires Pythagoras' theorem
- Right-angled triangles: Use the two perpendicular sides as base and height directly
Quadrilaterals
Different quadrilaterals require specific formulae:
Rectangle
- Area = length × width
- Perimeter = 2(length + width)
Square
- Area = side²
- Perimeter = 4 × side
Parallelogram
- Area = base × perpendicular height
- The slant height is not used for area calculations
- Perimeter = 2(a + b) where a and b are adjacent side lengths
Trapezium
- Area = ½(a + b)h where a and b are the parallel sides and h is the perpendicular distance between them
- This formula is given on the CIE formulae sheet, but you must identify a, b, and h correctly
Rhombus
- Area = base × height (like a parallelogram)
- Alternative: Area = ½ × d₁ × d₂ (using diagonals)
Kite
- Area = ½ × d₁ × d₂ where d₁ and d₂ are the lengths of the diagonals
For irregular quadrilaterals, split them into triangles and sum the areas.
Circles and parts of circles
Circle formulae (radius r, diameter d):
- Circumference = 2πr = πd
- Area = πr²
Sector formulae (angle θ in degrees, radius r):
- Arc length = (θ/360) × 2πr
- Sector area = (θ/360) × πr²
The fraction θ/360 represents what portion of the full circle the sector occupies.
Perimeter of a sector = arc length + 2r (don't forget the two radii)
Segment area = sector area − triangle area (when the chord creates a segment)
Exam questions frequently require answers in terms of π (exact form) or as decimal approximations to 3 significant figures. Read the question carefully to determine which format is required.
Compound shapes
Compound shapes combine multiple basic shapes. The systematic approach tested in exams:
- Identify the simple shapes that make up the compound figure
- Decompose by drawing lines to separate shapes clearly
- Calculate area or perimeter for each component
- Combine results appropriately:
- For area: add or subtract depending on whether shapes are added together or removed (holes)
- For perimeter: only count the outer boundary; internal divisions are not part of the perimeter
Common compound shapes in CIE papers:
- L-shapes and T-shapes: Split into rectangles
- Shapes with semicircular ends: Rectangle plus semicircle(s)
- Shaded regions: Subtract smaller shape area from larger shape area
- Rectilinear shapes: Shapes with all right angles; find missing lengths using given dimensions
When calculating perimeter of compound shapes with curves, remember that a semicircle contributes ½πd to the perimeter, not the diameter itself.
Problem-solving with units
Unit conversions appear regularly:
- 1 m = 100 cm, so 1 m² = 10,000 cm²
- 1 km = 1000 m, so 1 km² = 1,000,000 m²
- When converting area: square the conversion factor
- When converting volume: cube the conversion factor
Always check that your final answer uses the units requested in the question. Marks are lost for incorrect or missing units.
Using dimensions to check formulae
Dimensional analysis helps verify formulae:
- Perimeter formulae must combine lengths: [L]
- Area formulae must produce [L²]: length × length
- Volume formulae produce [L³]: length × length × length
This technique prevents errors like using slant height instead of perpendicular height in area calculations.
Worked examples
Example 1: Trapezium area and problem-solving
A trapezium has parallel sides of length 8 cm and 14 cm. The perpendicular distance between the parallel sides is 5 cm. Calculate: (a) the area of the trapezium (b) the area of a similar trapezium whose parallel sides are 12 cm and 21 cm
Solution:
(a) Using the formula: Area = ½(a + b)h
Area = ½(8 + 14) × 5 = ½ × 22 × 5 = 55 cm²
(b) The scale factor for lengths = 12/8 = 1.5 (or 21/14 = 1.5)
For similar shapes, area scale factor = (length scale factor)²
Area scale factor = 1.5² = 2.25
New area = 55 × 2.25 = 123.75 cm²
Example 2: Sector perimeter and area
A sector of a circle has radius 9 cm and angle 80°.
Calculate: (a) the arc length [2 marks] (b) the perimeter of the sector [1 mark] (c) the area of the sector, giving your answer in terms of π [2 marks]
Solution:
(a) Arc length = (θ/360) × 2πr
= (80/360) × 2π × 9 = (2/9) × 18π = 4π cm = 12.6 cm (3 s.f.) ✓✓
(b) Perimeter = arc length + 2r
= 12.6 + 2(9) = 30.6 cm ✓
(c) Sector area = (θ/360) × πr²
= (80/360) × π × 9² = (2/9) × 81π = 18π cm² ✓✓
Example 3: Compound shape with semicircles
The diagram shows a shape made from a rectangle 10 cm by 6 cm with semicircles on two opposite sides.
Calculate: (a) the perimeter of the shape [3 marks] (b) the area of the shape [3 marks]
Solution:
(a) The perimeter consists of two lengths of 10 cm and two semicircular arcs with diameter 6 cm.
Two semicircles = one full circle with diameter 6 cm
Perimeter = 2(10) + πd = 20 + π × 6 = 20 + 6π = 38.8 cm (3 s.f.) ✓✓✓
(b) Area = rectangle area + two semicircle areas
Rectangle area = 10 × 6 = 60 cm² ✓
Two semicircles = one circle with radius 3 cm
Circle area = πr² = π × 3² = 9π cm² ✓
Total area = 60 + 9π = 88.3 cm² (3 s.f.) ✓
Common mistakes and how to avoid them
• Using slant height instead of perpendicular height — In parallelograms, trapeziums, and triangles, only the perpendicular height works in area formulae. Always look for the right angle symbol or calculate the perpendicular distance using trigonometry.
• Forgetting to halve in triangle and trapezium formulae — The factor of ½ is essential. Write out the full formula first, then substitute values to avoid omitting this.
• Mixing up radius and diameter — Circumference = πd but Area = πr². Check whether the question gives radius or diameter before applying circle formulae. If given diameter, halve it to find radius for the area formula.
• Not including both radii when finding sector perimeter — Sector perimeter = arc length + 2r. Many students calculate only the arc length. The two straight edges (radii) form part of the boundary.
• Incorrectly combining areas in compound shapes — For shaded regions, determine whether to add or subtract. Draw clear diagrams showing which areas to calculate. If a shape is removed (like a circle cut from a rectangle), subtract its area.
• Unit conversion errors — When converting between m² and cm², multiply or divide by 10,000, not 100. The conversion factor must be squared because area is two-dimensional. Always write out the conversion explicitly: 1 m² = 100 cm × 100 cm = 10,000 cm².
Exam technique for perimeter and area of 2D shapes
• Command words matter: "Calculate" requires a numerical answer with working; "find" is similar but may be simpler; "show that" requires you to demonstrate a given answer, so work to more decimal places than shown; "give your answer in terms of π" means leave π in the answer (e.g., 18π cm²), not as a decimal.
• Mark allocation guides working: A [3 mark] question typically requires three steps—often: substitute into formula (1 mark), correct calculation process (1 mark), correct answer with units (1 mark). Show each step clearly on separate lines.
• Accuracy requirements: Unless specified otherwise, give answers to 3 significant figures. When the question specifies exact form (terms of π or surds), do not convert to decimals or you'll lose marks.
• Drawing on diagrams helps: For compound shapes, sketch division lines directly on the diagram to show how you split it. Label calculated lengths, especially missing dimensions found from given information. This organizes your thinking and can earn method marks even if arithmetic errors occur.
Quick revision summary
Triangles: Area = ½bh, perimeter = sum of sides. Rectangles: Area = lw, perimeter = 2(l+w). Parallelograms/Trapeziums: Use perpendicular height. Circles: A = πr², C = 2πr. Sectors: Multiply circle formulae by (θ/360). Compound shapes: Split into simple shapes, add/subtract areas, count only outer edges for perimeter. Convert units carefully—square the factor for area. Always include units in final answers and check whether exact form (with π) or decimal approximation is required.