Robinson Tuition
Year 8 · Key Stage 3 · free KS3 maths worksheet

Angles in polygons

12 original questions, 37 marks. Type your answers below for an instant mark and a score (algebra in any equivalent form, e.g. 2sqrt(3) or 2√3), or print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

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The questions

1

The diagrams show a triangle and a quadrilateral.

  1. Work out the size of angle x.
  2. Work out the size of angle y.
52°73°xDiagram NOT accurately drawna84°108°77°yDiagram NOT accurately drawnb
2 marks
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a) 55° b) 91°
2

A stop sign is the shape of a regular octagon.

  1. Work out the size of each exterior angle of a regular octagon.
  2. Work out the size of each interior angle of a regular octagon.
2 marks
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a) 45° b) 135°
3

Each exterior angle of a regular polygon is 30°.

Work out the number of sides of the polygon.

2 marks
Show answer
12
4

Work out the size of each interior angle of a regular decagon.

3 marks
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144°
5
  1. Work out the sum of the interior angles of a heptagon.
  2. The diagram shows a hexagon. Work out the size of angle x.
  3. Four of the exterior angles of a pentagon are 62°, 85°, 71° and 58°. Work out the size of the fifth exterior angle.
x135°128°95°142°104°Diagram NOT accurately drawn
4 marks
Show answer
a) 900° b) 116° c) 84°
6

DE and FG are parallel straight lines.

Triangle ABC is drawn so that A lies on DE, and B and C both lie on FG.

D is to the left of A and E is to the right of A.

Angle DAB = 48° and angle ACB = 63°.

Work out the size of angle BAC.

Give a reason for each stage of your working.

48°63°ABCDEFGDiagram NOT accurately drawn
4 marks
Show answer
69°
7

Each interior angle of a regular polygon is 156°.

Work out the number of sides of the polygon.

3 marks
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15
8

A pattern of shapes tessellates if the shapes fit together around every point with no gaps and no overlaps.

  1. Explain why regular hexagons tessellate.
  2. Explain why regular pentagons do not tessellate.
  3. Ella makes a floor tile pattern where two regular octagons and one square meet at every point. Show that this pattern has no gaps.
3 marks
Show answer
a) Each interior angle is 120°, and 3 × 120° = 360°, so three hexagons fit exactly round a point. b) Each interior angle is 108°; 3 × 108° = 324° leaves a gap and 4 × 108° = 432° overlaps, as 360 is not a multiple of 108. c) 135° + 135° + 90° = 360°.
9

The four angles of a quadrilateral are 74°, 101°, 2t° and (t + 5)°.

Work out the size of the largest angle of the quadrilateral.

3 marks
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120°
10

The interior angles of a hexagon are 2x°, 2x°, 3x°, 3x°, 4x° and 4x°.

Work out the size of the largest angle.

3 marks
Show answer
160°
11

The interior angles of a pentagon are x°, (x + 30)°, (x + 40)°, 2x° and (2x + 50)°.

Work out the size of the largest interior angle of the pentagon.

4 marks
Show answer
170°
12

ABCDE is a regular pentagon.

The diagonal AC is drawn.

Work out the size of angle ACD.

Give a reason for each stage of your working.

ABCDEDiagram NOT accurately drawn
4 marks
Show answer
72°

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Every question here is original and written by Robinson Tuition. It follows the Key Stage 3 programme of study; Year 9 sheets end with extension questions for top sets.

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