Regions in the Argand diagram

Reading the inequalities

• |z - a| < r: inside the circle; |z - a| > r: outside it.
• |z - a| < |z - b|: the side of the perpendicular bisector nearer to a.
• α < arg(z - a) < β: the sector (wedge) between two half-lines from a.

Use a solid boundary for ≤ or ≥ and a dashed boundary for < or >.

Combining conditions

Draw each boundary, decide which side satisfies each inequality, and shade only where all conditions hold.

Justify the side with a test point: the centre of a circle is usually easiest. The scheme of work quotes examiners noting that a surprising number of candidates shaded outside a circle instead of inside, and few used the centre to check.

Areas and extreme values

Areas are usually sectors, segments or parts of circles: a sector of angle θ (radians) has area ½r² θ.

The greatest argument in a disc comes from the tangent line drawn from the origin, so draw it and use right-angled triangle trigonometry.

Worked example

Shade the region where |z - 2 - 2i| ≤ 2 and |z| ≤ |z - 4|, and find its area.

1. |z - 2 - 2i| ≤ 2: inside or on the circle centre (2, 2), radius 2.
2. |z| ≤ |z - 4|: z is at least as near 0 as 4, so x ≤ 2 (the bisector of 0 and 4 is x = 2).
3. x = 2 passes through the centre, so the region is the left half of the disc.

Answer: Area = ½π(2)² = 2π

Check yourself

1. Describe the region |z - 3i| < |z + 1|.
   • Points nearer to -1 than to 3i
   • Points inside the circle centre 3i through -1
   • Points on the perpendicular bisector of 3i and -1
   • Points nearer to 3i than to -1
   Show answer

   Points nearer to 3i than to -1. |z - 3i| is the distance to 3i and |z + 1| is the distance to -1.

2. Find the area of the region 0 ≤ arg z ≤ π/3, |z| ≤ 6, to 2 decimal places.
   Show answer

   18.85. A sector of radius 6 and angle π/3: ½ × 36 × π/3 = 6π = 18.85.

3. Which point lies in the region |z - 2| < 1?
   • 1 + i
   • 2 + 2i
   • 0
   • 2.5 + 0.5i
   Show answer

   2.5 + 0.5i. |2.5 + 0.5i - 2| = |0.5 + 0.5i| = √0.5 ≈ 0.71 < 1. The others are √2, 2 and 2 away from the point 2.

4. Find the area of the region |z - 2 - 2i| ≤ 2, Re z ≤ 2, to 2 decimal places.
   Show answer

   6.28. Re z = 2 passes through the centre, so this is half the disc: ½π(2²) = 2π = 6.28.

5. When sketching |z + 1| < 3, how should the boundary circle be drawn?
   • As a solid line, because it is a circle
   • It should not be drawn at all
   • As a dashed line, because points on the circle are not included
   • As a solid line, because 3 is positive
   Show answer

   As a dashed line, because points on the circle are not included. A strict inequality excludes the boundary, which is shown by a dashed line.

6. z lies in the region |z - 4i| ≤ 2. The greatest value of arg z is kπ. Find k. Give your answer as a fraction in its simplest form.
   Show answer

   2/3. The tangent from the origin touches the circle where the radius is perpendicular to it. sin α = 2/4, so α = π/6 either side of the imaginary axis. Greatest argument = π/2 + π/6 = 2π/3.

7. z lies in the region |z| ≤ 2. Find the least value of |z - 6|.
   Show answer

   4. The point of the disc nearest to 6 is z = 2, which is 4 away.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit F2
• Pearson Edexcel IAL Further Pure Mathematics 2 scheme of work

Last updated 8 November 2026. Spotted a mistake? Tell us.