Circles, Arcs and Sectors
Mathematics (Core) revision notes · Cambridge (CIE) · 0580 · Core
1. Circumference and area
The two formulas
- The distance round a circle is its circumference, and the space inside is its area
- Both formulas are printed in the List of formulas at the front of the paper
- Area: A = πr²
- Circumference: C = 2πr, which is the same as C = πd because the diameter is twice the radius
- Read the diagram carefully to see whether the number given is the radius or the diameter, since halving or doubling at the wrong moment is the usual error
- Area needs the radius squared, so a diameter must be halved first
- Circumference is a length and takes units such as cm; area is a space and takes squared units such as cm²
Answers in terms of π
- Questions frequently ask for the answer "in terms of π", which means leaving π in the answer rather than multiplying it out
- Work in exact numbers throughout and collect the π at the end, so a circle of radius 5 has area 25π cm² and circumference 10π cm
- "In its simplest form" alongside it means the number in front of π should be fully simplified
- Only round when the question actually asks for a decimal, and then to whatever accuracy it states
Exam tip
Leave π in the final answer
When a question asks for an answer in terms of π, the last mark is only for an answer with π in it, such as 49π or 5π + 10. A decimal earns the method marks at most. Some mark schemes also count a correct π answer that you then change into a decimal as spoilt, so it loses the last mark as well. Stop once π is in place, unless a decimal is asked for too.
Worked example
Circumference and area of a circle
A circle has radius 11 cm. Find its circumference and its area, in terms of π and then to 3 significant figures.
Solution:
- Circumference = 2πr = 2 × π × 11 = 22π cm
- To 3 significant figures that is 69.1 cm
- Area = πr² = π × 11² = π × 121 = 121π cm²
- To 3 significant figures that is 380 cm²
- Square the radius before multiplying by π: (π × 11)² would be a different number entirely
- The circumference is a length, so its units are cm; the area is in cm²
2. Arcs and sectors
The vocabulary
- An arc is part of the circumference, and a sector is the slice enclosed by two radii and an arc
- The angle at the centre between the two radii is usually labelled θ
The two formulas
- An arc and a sector are simply fractions of the whole circle, and the fraction is θ ÷ 360
- Arc length = (θ ÷ 360) × 2πr
- Sector area = (θ ÷ 360) × πr²
- Neither formula is printed in the paper, so both have to be known, although each is just the circle formula scaled by the fraction
- θ is kept to a factor of 360°, such as 90°, 60° or 45°, so the sector is always a simple fraction of the whole circle
- To go all the way round a sector you travel along the arc and back down both radii, so its perimeter includes those two straight edges — routinely forgotten
Worked example
Arc, area and perimeter of a sector
A sector of a circle has radius 12 cm and sector angle 45°. Find its arc length, its area, and its perimeter. Give the first two in terms of π.
Solution:
- The fraction of the circle is 45 ÷ 360 = 1/8
- The full circumference is 2π × 12 = 24π cm
- Arc length = 1/8 × 24π = 3π cm
- The full area is π × 12² = 144π cm²
- Sector area = 1/8 × 144π = 18π cm²
- The perimeter is the arc plus both radii: 3π + 12 + 12
- Perimeter = (3π + 24) cm
3. Semicircles and compound shapes
Parts of circles
- A semicircle is half a circle, so work out the whole circle's area and take half of it
- Its perimeter is not half the circumference: it is the curved half plus the straight diameter across the bottom
- A semicircle of radius 6 cm has area 18π cm² and perimeter (6π + 12) cm
- A quarter circle works the same way, with two radii forming the straight edges
- Whenever a shape has a curved edge and a straight one, list the pieces of the perimeter before adding, so none is left out
Shapes built from circles
- Compound shapes combine circles or parts of circles with rectangles and triangles
- Add the pieces where the shape is built up, and subtract where a piece has been removed
- A shaded region between two circles with the same centre is the larger area minus the smaller
- Sketch the split and label each piece, since the separate areas carry the method marks
- Keep every part in terms of π until the end, which makes the final collecting much easier
Worked example
A semicircle's area and perimeter
A semicircle has diameter 12 cm. Find its area and its perimeter, in terms of π.
Solution:
- The radius is half the diameter, so r = 6 cm
- For the area, find the whole circle's area and halve it
- Whole circle: π × 6² = 36π
- Area of the semicircle = 36π ÷ 2 = 18π cm²
- The perimeter is not half the circumference: the straight edge counts too
- Curved part = half of 2π × 6 = 6π
- Straight part = the diameter = 12
- Perimeter = (6π + 12) cm
- Stopping at 6π is the standard error: it leaves out the 12 cm straight edge
Worked example
A rectangle with a semicircle cut out
A metal plate is a rectangle 15 cm long and 8 cm wide, with a semicircle of diameter 8 cm cut out of one of its short ends. Find the area and the perimeter of the plate, each correct to 3 significant figures.
Solution:
- Split the plate into pieces: the whole rectangle, with a semicircle taken away
- Rectangle = 15 × 8 = 120 cm²
- The semicircle's radius is 8 ÷ 2 = 4 cm, so its area is half of π × 4², which is 8π cm²
- Area of the plate = 120 − 8π = 94.86… = 94.9 cm²
- For the perimeter, list every edge round the outside: two long sides, the straight short end, and the curved edge of the cut
- The 8 cm straight edge at the cut end is no longer on the outside: the curved edge has replaced it
- Curved edge = half of 2π × 4 = 4π cm
- Perimeter = 15 + 15 + 8 + 4π = 50.56… = 50.6 cm