Hard GCSE Angles in polygons Questions

Challenging, exam-style GCSE Angles in polygons questions with worked solutions. Stretch yourself on the hardest regular polygon, equal angles, interior to exterior, ratio problems.

regular polygonequal anglesinterior to exteriorratioforming an equationinterior angle sum
GCSE Foundation34 questionsStep-by-step solutions
Question 1
5 markschallenging
Each interior angle of a regular polygon is 88 times the size of each exterior angle. Which statement is correct?
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Worked solution

  1. Use the fact that an interior angle and its exterior angle lie on a straight line.

    i+e=180∘i + e = 180^\circ

    At every vertex the interior angle and the exterior angle make a straight line, so they add to 180∘180^\circ.

  2. Write the interior angle in terms of the exterior angle.

    i=8ei = 8e

    The interior angle is 88 times the exterior angle.

  3. Substitute into the straight line fact.

    8e+e=180∘8e + e = 180^\circ

    This leaves an equation in ee alone.

  4. Collect the terms.

    9e=180∘9e = 180^\circ

    8e+e=9e8e + e = 9e.

  5. Divide to find the exterior angle.

    e=180∘9=20∘e = \frac{180^\circ}{9} = 20^\circ

    180÷9=20180 \div 9 = 20.

  6. Recall that the exterior angles of any polygon add up to 360 degrees.

    sum of exterior angles=360∘\text{sum of exterior angles} = 360^\circ

    Going once round the outside of a polygon turns you through one complete turn, so the exterior angles always total 360∘360^\circ, whatever the number of sides.

  7. Divide 360 degrees by the exterior angle.

    n=360∘20∘=18n = \frac{360^\circ}{20^\circ} = 18

    The polygon has 1818 sides.

  8. Find the interior angle.

    i=180∘−20∘=160∘i = 180^\circ - 20^\circ = 160^\circ

    Each interior angle is 160∘160^\circ.

  9. Check the multiple asked for.

    16020=8\frac{160}{20} = 8

    The interior angle really is 88 times the exterior angle.

  10. Test the other options in turn.

    9→14040=3.5,16→157.522.5=7,10→14436=4,20→16218=99 \rightarrow \frac{140}{40} = 3.5, \quad 16 \rightarrow \frac{157.5}{22.5} = 7, \quad 10 \rightarrow \frac{144}{36} = 4, \quad 20 \rightarrow \frac{162}{18} = 9

    For each of the other numbers of sides the interior angle is not 88 times the exterior angle.

  11. Check the exterior angles of the whole polygon add to 360 degrees.

    18×20∘=360∘18 \times 20^\circ = 360^\circ

    The 18 equal exterior angles come to 360∘360^\circ, as they must.

  12. Check the answer against the interior angle sum.

    18×160∘=2880∘=(18−2)×180∘18 \times 160^\circ = 2880^\circ = (18 - 2) \times 180^\circ

    The 18 equal interior angles total 2880∘2880^\circ, which is exactly (18−2)×180∘(18 - 2) \times 180^\circ.

  13. Check the answer is sensible.

    160∘<180∘160^\circ < 180^\circ

    Each interior angle of a convex polygon is less than 180∘180^\circ, and 160∘160^\circ is.

  14. Note the general result.

    n=360∘180∘8+1=2(8+1)n = \frac{360^\circ}{\frac{180^\circ}{8 + 1}} = 2(8 + 1)

    Whenever the interior angle is 88 times the exterior angle, the polygon has 2×(8+1)=182 \times (8 + 1) = 18 sides.

  15. Select the correct statement.

    n=18n = 18

    The polygon has 1818 sides.

Answer
n=18n = 18
Question 2
5 markschallenging
A regular polygon has 1212 sides. Which statement is correct?
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Worked solution

  1. Recall that the exterior angles of any polygon add up to 360 degrees.

    sum of exterior angles=360∘\text{sum of exterior angles} = 360^\circ

    Going once round the outside of a polygon turns you through one complete turn, so the exterior angles always total 360∘360^\circ, whatever the number of sides.

  2. Find one exterior angle.

    e=360∘12=30∘e = \frac{360^\circ}{12} = 30^\circ

    The 1212 exterior angles are equal, so each is 360÷12=30360 \div 12 = 30 degrees.

  3. Use the fact that an interior angle and its exterior angle lie on a straight line.

    i+e=180∘i + e = 180^\circ

    At every vertex the interior angle and the exterior angle make a straight line, so they add to 180∘180^\circ.

  4. Find one interior angle.

    i=180∘−30∘=150∘i = 180^\circ - 30^\circ = 150^\circ

    Each interior angle is 150∘150^\circ.

  5. Check the answer against the interior angle sum.

    12×150∘=1800∘=(12−2)×180∘12 \times 150^\circ = 1800^\circ = (12 - 2) \times 180^\circ

    The 12 equal interior angles total 1800∘1800^\circ, which is exactly (12−2)×180∘(12 - 2) \times 180^\circ.

  6. Check the exterior angles of the whole polygon add to 360 degrees.

    12×30∘=360∘12 \times 30^\circ = 360^\circ

    The 12 equal exterior angles come to 360∘360^\circ, as they must.

  7. Check the answer is sensible.

    150∘<180∘150^\circ < 180^\circ

    Each interior angle of a convex polygon is less than 180∘180^\circ, and 150∘150^\circ is.

  8. Reject the swapped option.

    150∘≠e,30∘≠i150^\circ \ne e, \quad 30^\circ \ne i

    The interior angle is the LARGER of the two here, so swapping them over is wrong.

  9. Reject the option whose two angles do not make a straight line.

    30∘+160∘=190∘≠180∘30^\circ + 160^\circ = 190^\circ \ne 180^\circ

    The interior and exterior angles at a vertex must add to 180∘180^\circ, and these do not.

  10. Reject the decagon option.

    360∘10=36∘≠30∘\frac{360^\circ}{10} = 36^\circ \ne 30^\circ

    Those are the angles of a regular polygon with 1010 sides, not 1212.

  11. Reject the hexagon option.

    360∘6=60∘≠30∘\frac{360^\circ}{6} = 60^\circ \ne 30^\circ

    Those are the angles of a regular hexagon.

  12. Check the interior angle another way.

    (12−2)×180∘12=150∘\frac{(12 - 2) \times 180^\circ}{12} = 150^\circ

    Dividing the interior angle sum by 1212 gives the same interior angle.

  13. Note the pair that must be true.

    e=30∘,i=150∘e = 30^\circ, \quad i = 150^\circ

    Only one option has BOTH angles right.

  14. Check the two angles add to 180 degrees.

    30+150=18030 + 150 = 180

    The correct pair does lie on a straight line, as it must.

  15. Select the correct statement.

    e=30∘, i=150∘e = 30^\circ, \ i = 150^\circ

    Each exterior angle is 30∘30^\circ and each interior angle is 150∘150^\circ.

Answer
e=30∘, i=150∘e = 30^\circ, \ i = 150^\circ
Question 3
5 markschallenging
The diagram shows a regular pentagon and a regular decagon that share a common edge. Work out the size of the angle marked xx.
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Worked solution

  1. Find one interior angle of the regular pentagon.

    i1=180∘−360∘5=108∘i_1 = 180^\circ - \frac{360^\circ}{5} = 108^\circ

    Its exterior angle is 360÷5=72360 \div 5 = 72 degrees, so its interior angle is 108∘108^\circ.

  2. Find one interior angle of the regular decagon.

    i2=180∘−360∘10=144∘i_2 = 180^\circ - \frac{360^\circ}{10} = 144^\circ

    Its exterior angle is 360÷10=36360 \div 10 = 36 degrees, so its interior angle is 144∘144^\circ.

  3. Use the fact that angles at a point add to 360 degrees.

    i1+i2+x=360∘i_1 + i_2 + x = 360^\circ

    The two interior angles and the angle xx together make a full turn at the shared vertex.

  4. Add the two interior angles.

    108+144=252108 + 144 = 252

    Together the two shapes take up 252∘252^\circ of the full turn.

  5. Subtract from 360 degrees.

    x=360−252=108x = 360 - 252 = 108

    The gap left is 108∘108^\circ.

  6. Check the three angles add to a full turn.

    108+144+108=360108 + 144 + 108 = 360

    The three angles at the shared vertex total 360∘360^\circ, as they must.

  7. Check the first interior angle another way.

    (5−2)×180∘5=108∘\frac{(5 - 2) \times 180^\circ}{5} = 108^\circ

    Dividing the interior angle sum of the regular pentagon by 55 gives the same interior angle.

  8. Check the second interior angle another way.

    (10−2)×180∘10=144∘\frac{(10 - 2) \times 180^\circ}{10} = 144^\circ

    Dividing the interior angle sum of the regular decagon by 1010 gives the same interior angle.

  9. Note why the shapes leave a gap.

    108+144≠360108 + 144 \ne 360

    The two interior angles do not fill the turn, so the two polygons cannot meet a third copy without a gap.

  10. Check the answer is a sensible size.

    0∘<108∘<360∘0^\circ < 108^\circ < 360^\circ

    The gap is 108∘108^\circ, less than a full turn.

  11. Record the exterior angles used.

    72∘,36∘72^\circ, \quad 36^\circ

    These came from dividing 360∘360^\circ by the number of sides of each regular polygon.

  12. Check the answer against the exterior angles.

    72+36=10872 + 36 = 108

    Taking two interior angles from 360∘360^\circ leaves exactly the two exterior angles, so xx must be their sum — and it is.

  13. Record the interior angle sums of the two polygons.

    (5−2)×180∘=540∘,(10−2)×180∘=1440∘(5 - 2) \times 180^\circ = 540^\circ, \quad (10 - 2) \times 180^\circ = 1440^\circ

    These are the totals the two interior angles were taken from.

  14. Count the angles meeting at the shared vertex.

    3 angles:108∘, 144∘, 108∘3 \text{ angles}: 108^\circ, \ 144^\circ, \ 108^\circ

    Exactly three angles meet at that vertex, and together they make one complete turn.

  15. State the size of the angle marked x.

    x=108∘x = 108^\circ

    The angle marked xx is 108∘108^\circ.

Answer
x=108∘x = 108^\circ
Question 4
6 markschallenging
The diagram shows an octagon. Its interior angles are x∘x^\circ, (x+15)∘(x + 15)^\circ, (x+25)∘(x + 25)^\circ, (x+35)∘(x + 35)^\circ, (x+40)∘(x + 40)^\circ, (x+50)∘(x + 50)^\circ, (x+55)∘(x + 55)^\circ and (x+60)∘(x + 60)^\circ. Work out the size of the largest interior angle.
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Worked solution

  1. Work out the sum of the interior angles of the polygon.

    S=(8−2)×180∘=1080∘S = (8 - 2) \times 180^\circ = 1080^\circ

    An octagon has 88 sides, so its interior angles total 1080∘1080^\circ.

  2. Add the expressions for the angles.

    x+(x+15)+(x+25)+(x+35)+(x+40)+(x+50)+(x+55)+(x+60)=1080x + (x + 15) + (x + 25) + (x + 35) + (x + 40) + (x + 50) + (x + 55) + (x + 60) = 1080

    The interior angles must add up to the interior angle sum.

  3. Collect the x terms.

    8x+280=10808x + 280 = 1080

    There are 88 lots of xx altogether, and the numbers add to 280280.

  4. Subtract the number term from both sides.

    8x=1080−280=8008x = 1080 - 280 = 800

    1080−280=8001080 - 280 = 800.

  5. Divide both sides by the coefficient of x.

    x=8008=100x = \frac{800}{8} = 100

    800÷8=100800 \div 8 = 100.

  6. Substitute back to find every angle.

    100,115,125,135,140,150,155,160100, 115, 125, 135, 140, 150, 155, 160

    Putting x=100x = 100 into each expression gives the angles, in degrees.

  7. Check the angles add to the interior angle sum.

    100+115+125+135+140+150+155+160=1080100 + 115 + 125 + 135 + 140 + 150 + 155 + 160 = 1080

    They total 1080∘1080^\circ, so the value of xx is correct.

  8. Check every angle is less than 180 degrees.

    max⁡=160∘<180∘\max = 160^\circ < 180^\circ

    All the angles are under 180∘180^\circ, so the polygon really is convex.

  9. Identify the expression that gives the largest angle.

    (x+60)∘=160∘(x + 60)^\circ = 160^\circ

    Evaluating the largest expression at x=100x = 100 gives 160∘160^\circ.

  10. Identify the smallest angle for comparison.

    min⁡=100∘\min = 100^\circ

    The smallest angle is 100∘100^\circ, so the angles run from 100∘100^\circ to 160∘160^\circ.

  11. Work out the exterior angles.

    80,65,55,45,40,30,25,2080, 65, 55, 45, 40, 30, 25, 20

    Each exterior angle is 180∘180^\circ minus the interior angle at that vertex.

  12. Check the exterior angles add to 360 degrees.

    80+65+55+45+40+30+25+20=36080 + 65 + 55 + 45 + 40 + 30 + 25 + 20 = 360

    The exterior angles of any polygon total 360∘360^\circ — a completely independent check on all the angles at once.

  13. Sanity check with the mean angle.

    10808=135\frac{1080}{8} = 135

    The angles average 135∘135^\circ, which lies between the smallest and the largest, as it must.

  14. Record the value of x.

    x=100x = 100

    The unknown in the expressions is x=100x = 100.

  15. State the largest interior angle.

    largest angle=160∘\text{largest angle} = 160^\circ

    The largest interior angle is 160∘160^\circ.

Answer
largest angle=160∘\text{largest angle} = 160^\circ
Question 5
6 markschallenging
The diagram shows a heptagon. Its interior angles are x∘x^\circ, (x+10)∘(x + 10)^\circ, (x+20)∘(x + 20)^\circ, (x+30)∘(x + 30)^\circ, (x+40)∘(x + 40)^\circ, (x+45)∘(x + 45)^\circ and (x+55)∘(x + 55)^\circ. Work out the size of the largest interior angle.
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Worked solution

  1. Work out the sum of the interior angles of the polygon.

    S=(7−2)×180∘=900∘S = (7 - 2) \times 180^\circ = 900^\circ

    A heptagon has 77 sides, so its interior angles total 900∘900^\circ.

  2. Add the expressions for the angles.

    x+(x+10)+(x+20)+(x+30)+(x+40)+(x+45)+(x+55)=900x + (x + 10) + (x + 20) + (x + 30) + (x + 40) + (x + 45) + (x + 55) = 900

    The interior angles must add up to the interior angle sum.

  3. Collect the x terms.

    7x+200=9007x + 200 = 900

    There are 77 lots of xx altogether, and the numbers add to 200200.

  4. Subtract the number term from both sides.

    7x=900−200=7007x = 900 - 200 = 700

    900−200=700900 - 200 = 700.

  5. Divide both sides by the coefficient of x.

    x=7007=100x = \frac{700}{7} = 100

    700÷7=100700 \div 7 = 100.

  6. Substitute back to find every angle.

    100,110,120,130,140,145,155100, 110, 120, 130, 140, 145, 155

    Putting x=100x = 100 into each expression gives the angles, in degrees.

  7. Check the angles add to the interior angle sum.

    100+110+120+130+140+145+155=900100 + 110 + 120 + 130 + 140 + 145 + 155 = 900

    They total 900∘900^\circ, so the value of xx is correct.

  8. Check every angle is less than 180 degrees.

    max⁡=155∘<180∘\max = 155^\circ < 180^\circ

    All the angles are under 180∘180^\circ, so the polygon really is convex.

  9. Identify the expression that gives the largest angle.

    (x+55)∘=155∘(x + 55)^\circ = 155^\circ

    Evaluating the largest expression at x=100x = 100 gives 155∘155^\circ.

  10. Identify the smallest angle for comparison.

    min⁡=100∘\min = 100^\circ

    The smallest angle is 100∘100^\circ, so the angles run from 100∘100^\circ to 155∘155^\circ.

  11. Work out the exterior angles.

    80,70,60,50,40,35,2580, 70, 60, 50, 40, 35, 25

    Each exterior angle is 180∘180^\circ minus the interior angle at that vertex.

  12. Check the exterior angles add to 360 degrees.

    80+70+60+50+40+35+25=36080 + 70 + 60 + 50 + 40 + 35 + 25 = 360

    The exterior angles of any polygon total 360∘360^\circ — a completely independent check on all the angles at once.

  13. Sanity check with the mean angle.

    9007=9007\frac{900}{7} = \frac{900}{7}

    The angles average 9007∘\frac{900}{7}^\circ, which lies between the smallest and the largest, as it must.

  14. Record the value of x.

    x=100x = 100

    The unknown in the expressions is x=100x = 100.

  15. State the largest interior angle.

    largest angle=155∘\text{largest angle} = 155^\circ

    The largest interior angle is 155∘155^\circ.

Answer
largest angle=155∘\text{largest angle} = 155^\circ

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