AQA GCSE Mathematics Paper 2 (Higher), June 2025: Question 11

3 marks · Medium difficulty · Multi-step Problem

Work out the size of angle x formed by a diagonal connecting opposite vertices of a regular octagon.

Practise this question

Question

A regular octagon with horizontal top and bottom sides, and vertical left and right sides. A straight line diagonal is drawn across the octagon connecting the vertex between the left vertical side and the bottom-left slanted side to the opposite vertex between the top-right slanted side and the right vertical side. Angle x is marked between the left vertical side and the drawn straight line. The question asks to work out the size of angle x.

Mark scheme

Show the mark scheme Mark scheme table for question 11. Answer: 67.5 with mark B3 (or equivalent value, e.g. 135/2). Partial marks: B2 for finding an interior angle of 135 degrees; B1 for finding an exterior angle of 45 degrees or total interior angles of 1080 degrees. Additional guidance notes that 67.5 must be their answer for B3.

How to answer it

Angles in a Regular Octagon: Line of Symmetry

What This Question Tests

This question assesses your ability to calculate interior and exterior angles of regular polygons and recognise lines of symmetry within 2D geometric shapes. You must deduce that a line passing directly through opposite vertices of an 8-sided polygon bisects the interior angle.

Question 11 Walkthrough

A straight line is drawn across a regular octagon. Work out the size of angle x. [3 marks]

💡 Understanding the Diagram & Symmetry

  • An octagon has 8 sides and 8 vertices. Because it is regular, all sides are equal and all interior angles are identical.
  • Counting the edges: there are exactly 4 sides above the straight line and 4 sides below it.
  • This means the drawn line runs directly through the centre between two opposite vertices, forming a line of reflective symmetry.
  • Consequently, this line divides the interior angle at the vertex exactly in half ( x = interior angle ÷ 2 ).

📐 Step-by-Step Calculation

Method 1: Using the Exterior Angle (Fastest & Safest)

1. All exterior angles of any convex polygon add to 360°.
    Exterior angle = 360° ÷ 8 = 45°   (Awards 1st mark: B1)
2. Interior angle + Exterior angle = 180° (angles on a straight line):
    Interior angle = 180° - 45° = 135°   (Awards 2nd mark: B2)
3. The diagonal line bisects the interior angle:
     x = 135° ÷ 2 = 67.5°   (Awards final mark: B3)

Method 2: Using the Sum of Interior Angles Formula

1. Formula for sum of interior angles: (n - 2) × 180°
    Sum = (8 - 2) × 180° = 6 × 180° = 1080°   (Awards 1st mark: B1)
2. Each interior angle in a regular octagon:
    Interior angle = 1080° ÷ 8 = 135°   (Awards 2nd mark: B2)
3. Bisect the angle:
     x = 135° ÷ 2 = 67.5°   (Awards final mark: B3)

✅ Final Answer

x = 67.5°

Also accepted: equivalent values such as 135/2 or 67 ½° .

🧠 Exam Technique & Examiner Insight

  • Show intermediate values: Writing down 45° , 1080° , or 135° guarantees method marks even if an arithmetic slip occurs at the end.
  • Sense check visually: The angle x is acute (less than 90°). An answer like 135° is obviously too large for x .

❌ Common Errors to Avoid

  • Stopping at 135°: Many candidates correctly found the interior angle of an octagon ( 135° ) but forgot to divide by 2, missing the final answer mark.
  • Confusing interior and exterior angles: Dividing 360° ÷ 8 = 45° and incorrectly assuming 45° is the interior angle, leading to 45 ÷ 2 = 22.5° .
  • Miscounting sides: Calculating angles for a hexagon (6 sides) or decagon (10 sides) instead of an octagon (8 sides).
Mark Scheme Breakdown:
  • B1: Finding exterior angle 45° OR sum of interior angles 1080° .
  • B2: Finding regular octagon interior angle 135° .
  • B3: Fully correct answer of 67.5° (or equivalent fraction).

Topics

Geometry and measures · 3.4.1 Properties and constructions

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 2 (Higher), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.