Angles in polygons – Year 9 maths test answers to download as PDF

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1. Answer to Question 1 (4 marks)

   1. Use the rule sum \(=(n-2)\times180°\) with \(n=6\): \((6-2)\times180=\)\(720°\) (1 mark). Drawing diagonals from one vertex splits a hexagon into 4 triangles, so its angle sum is \(4\times180°\).
   2. Find the exterior angle first by sharing the full turn between 8 equal angles: \(360\div8=\)\(45°\) (1 mark). The interior angle sits on a straight line with it: \(180-45=\)\(135°\) (1 mark).
   3. The exterior angles of any polygon add up to 360°, so the number of sides is \(360\div24=\)15 (1 mark). The 15 equal exterior angles fill the full turn.

   Check (b): \(8\times135=1080=(8-2)\times180\). Each interior angle of the 15-sided polygon is \(180-24=156°\).

2. Answer to Question 2 (4 marks)

   1. A pentagon splits into \(5-2=3\) triangles, so its interior angles add up to \(3\times180=\)\(540°\). (1 mark)
   2. The five angles must add up to 540°, and the two unknown angles are both x: \(120+100+110+x+x=540\), which simplifies to \(330+2x=540\). (1 mark)
   3. Subtract 330 from both sides: \(2x=210\) (1 mark). Divide both sides by 2: \(x=\)\(105°\) (1 mark).

   Check: \(120+100+110+105+105=540\). Every angle is less than 180°, so the pentagon is convex, which is sensible. A common error is writing \(x\) only once in the equation, which would give \(x=210°\), an impossible reflex angle here.

3. Answer to Question 3 (4 marks)

   1. A hexagon has angle sum 720°, so each interior angle of a regular hexagon is \(720\div6=120°\) (1 mark). Three hexagons meeting at a point give \(3\times120=360°\), exactly a full turn, so they fit together with no gap and no overlap (1 mark for clearly stating the total is 360°).
   2. The exterior angle of a regular decagon is \(360\div10=36°\), so the interior angle is \(180-36=144°\) (1 mark). Then \(360\div144=2.5\), not a whole number, so a whole number of decagons cannot meet at a point (1 mark).

   Two decagons leave a gap of \(360-288=72°\), which is too small for a third decagon. This is why honeycombs use hexagons: only triangles, squares and hexagons tile on their own.

4. Answer to Question 4 (4 marks)

   1. Each turn of 36° is an exterior angle of the path, and the turns must add up to one full turn of 360° for the robot to face its starting direction again. It repeats the command \(360\div36=\)10 times. (1 mark)
   2. At each corner the interior and exterior angles lie on a straight line: \(180-36=\)\(144°\). (1 mark)
   3. The path is a regular decagon with 10 sides of 1.5 m, so the robot travels \(10\times1.5=\)15 m. (1 mark)
   4. With 45° turns: \(360\div45=8\) equal sides, so it traces a regular octagon. (1 mark)

   The octagon path would be \(8\times1.5=12\) m long.

5. Answer to Question 5 (4 marks)

   1. At each vertex the interior and exterior angles make a straight line, so the exterior angle is \(180-130=\)\(50°\). (1 mark)
   2. For a regular polygon the number of sides is \(360\div\) exterior angle \(=360\div50=7.2\) (1 mark). A polygon must have a whole number of sides, so no regular polygon has an interior angle of 130° and Jake cannot be correct (1 mark).
   3. Exterior angle \(=180-135=45°\), so the number of sides is \(360\div45=\)8; it is a regular octagon. (1 mark)

   The regular heptagon (7 sides) has interior angles of about 128.6° and the octagon 135°, so 130° falls between them. Only exterior angles that divide exactly into 360 give real regular polygons.

Total : 20 marks