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Complete IGCSE Edexcel guide to circle formulae, arc length, sector area, segment area and compound shapes. Worked examples for grades 5-8.
Circumference = pi times d (or 2 pi r). Area = pi r squared. For a sector with angle theta at the centre: arc length = (theta / 360) times 2 pi r, and sector area = (theta / 360) times pi r squared. For a segment: segment area = sector area minus triangle area. These formulae cover every standard circle mensuration question on the IGCSE Edexcel exam.
Source: Edexcel IGCSE Mathematics (9-1) Specification 4MA1
The two fundamental circle formulae are the starting point for every mensuration question involving circles, sectors, and segments.
The circumference (perimeter) of a circle is the total distance around its edge. There are two equivalent formulae:
C = pi x d (pi times the diameter)
C = 2 x pi x r (2 times pi times the radius)
Since the diameter is twice the radius (d = 2r), both formulae are identical. Choose whichever matches the information given in the question.
The area of a circle is the space enclosed within it:
The radius must be squared. If you are given the diameter, halve it first to get the radius before substituting into the formula. This is the most common source of error.
A sector is a "slice" of a circle, bounded by two radii and an arc. The angle at the centre (theta) determines what fraction of the full circle the sector represents.
| Measurement | Full Circle | Sector (angle theta) |
|---|---|---|
| Length | C = 2 pi r | Arc = (theta / 360) x 2 pi r |
| Area | A = pi r squared | Sector = (theta / 360) x pi r squared |
The fraction theta / 360 is the proportion of the full circle. For example, a 90-degree sector is 90/360 = 1/4 of the circle, so its arc length is a quarter of the circumference and its area is a quarter of the full area.
The perimeter of a sector includes the arc length plus the two radii: perimeter = arc length + 2r.
A segment is the region between a chord and the arc it cuts off. To find its area, subtract the triangle area from the sector area.
Segment area = Sector area - Triangle area
Sector area = (theta / 360) x pi r squared
Triangle area = (1/2) x r squared x sin(theta)
The triangle is formed by the two radii and the chord. Its area is calculated using the formula (1/2)ab sin C, where a and b are both equal to the radius and C is the angle between them (theta). This simplifies to (1/2) r squared sin(theta).
Segment questions typically appear at Grade 7-8 and are a favourite for the higher end of the paper because they combine sector knowledge with trigonometry.
Compound shape questions ask you to find the area or perimeter of a shape made from combinations of circles, semicircles, sectors, rectangles and triangles.
The strategy is always the same:
For perimeters, only count edges that form the outer boundary. Internal edges where shapes meet are not part of the perimeter.
🎯 A circle has radius 7 cm. Find its circumference and area. Give answers to 1 decimal place.
Step 1 (Circumference): C = 2 x pi x r = 2 x pi x 7 = 14 pi
Step 2 (Evaluate): C = 14 x 3.14159... = 43.982... = 44.0 cm (1 d.p.)
Step 3 (Area): A = pi x r squared = pi x 7 squared = pi x 49 = 49 pi
Step 4 (Evaluate): A = 49 x 3.14159... = 153.938... = 153.9 cm squared (1 d.p.)
Answer: Circumference = 44.0 cm, Area = 153.9 cm squared
🎯 A sector has radius 10 cm and an angle of 120 degrees at the centre. Find the arc length and the sector area. Give answers to 3 significant figures.
Step 1 (Fraction of circle): theta / 360 = 120 / 360 = 1/3
Step 2 (Arc length): Arc = (1/3) x 2 x pi x 10 = (1/3) x 20 pi = 20 pi / 3
Step 3 (Evaluate arc): = 20.944... = 20.9 cm (3 s.f.)
Step 4 (Sector area): Sector = (1/3) x pi x 10 squared = (1/3) x 100 pi = 100 pi / 3
Step 5 (Evaluate area): = 104.719... = 105 cm squared (3 s.f.)
Answer: Arc length = 20.9 cm, Sector area = 105 cm squared
🎯 A circle has radius 12 cm. A chord subtends an angle of 80 degrees at the centre. Find the area of the minor segment. Give your answer to 3 significant figures.
Step 1 (Sector area): Sector = (80 / 360) x pi x 12 squared = (80 / 360) x 144 pi = (2/9) x 144 pi = 288 pi / 9 = 32 pi
Step 2 (Evaluate sector): 32 pi = 100.531 cm squared
Step 3 (Triangle area): Triangle = (1/2) x r squared x sin(theta) = (1/2) x 144 x sin(80 degrees)
Step 4 (Evaluate triangle): = 72 x sin(80 degrees) = 72 x 0.98481 = 70.906 cm squared
Step 5 (Segment area): Segment = Sector - Triangle = 100.531 - 70.906 = 29.625 cm squared
Step 6 (Round): = 29.6 cm squared (3 s.f.)
Answer: 29.6 cm squared
Using diameter instead of radius: The area formula requires the radius. If the question gives the diameter (e.g. 14 cm), you must halve it (r = 7 cm) before substituting. Using 14 in A = pi r squared gives an answer four times too large.
Forgetting to square the radius: A = pi x r squared, not pi x r. This is the most common arithmetic error. Writing pi x 7 = 21.99 instead of pi x 49 = 153.94 loses all marks.
Rounding too early: In multi-step problems (especially segments), keep at least 4 decimal places in intermediate steps. Round only at the very end to the accuracy stated in the question.
Mixing up arc length and sector area: Arc length is a measure of distance (cm). Sector area is a measure of space (cm squared). Make sure you are answering the right question.
Check radius vs diameter first: Before substituting anything, identify whether you have been given the radius or the diameter. Write "r = ..." clearly at the top of your working.
Leave in terms of pi when possible: If the question says "give your answer in terms of pi", do not evaluate to a decimal. Write your answer as a multiple of pi (e.g. 49 pi cm squared).
Show the fraction theta/360: Always write out the fraction of the circle you are using. This earns method marks and helps you avoid errors. For a 120-degree sector, write 120/360 = 1/3.
Sketch compound shapes: For complex problems, sketch the shape and label each part. Identify which areas to add and which to subtract before calculating anything.
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