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Geometry and proof · Ages 13–14

Drawing a regular pentagon worksheet

Part of: Polygons

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Drawing a regular pentagon worksheet

Geometry and proof · Ages 13–14

Numsprite
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Name:Date:Score:／48

Write each answer in the space. For word problems, show your working and write a sentence answer.

Part A: Working out and filling in (2 marks each)

  1. 1.O is the centre of the regular pentagon. x =

    Answer:

  2. 2.The picture shows a regular pentagon. x =

    Answer:

  3. 3.ABCDE is a regular pentagon and AC a diagonal. ∠BAC =

    Answer:

  4. 4.Draw a regular pentagon in a circle of radius 5 cm with compasses, a ruler and a set square: put the steps in order.

    • A. MN = MC: with centre M and radius MC, draw an arc cutting OA at N
    • B. ∠AOC = 90°: draw a circle, centre O and radius 5 cm, and with a set square two diameters AB and CD at right angles
    • C. OM = MB: measure OB with the ruler to find its midpoint M
    • D. side = CN: with centre C and radius CN, mark off round the circle five times and join the points in turn

    Write the letters in the right order

    Answer:

  5. 5.Draw a regular pentagon in a circle of radius 6 cm with compasses, a ruler and a set square: put the steps in order.

    • A. OM = MB: measure OB with the ruler to find its midpoint M
    • B. MN = MC: with centre M and radius MC, draw an arc cutting OA at N
    • C. ∠AOC = 90°: draw a circle, centre O and radius 6 cm, and with a set square two diameters AB and CD at right angles
    • D. side = CN: with centre C and radius CN, mark off round the circle five times and join the points in turn

    Write the letters in the right order

    Answer:

  6. 6.Draw a regular pentagon in a circle of radius 3 cm with compasses, a ruler and a set square: put the steps in order.

    • A. MN = MC: with centre M and radius MC, draw an arc cutting OA at N
    • B. OM = MB: measure OB with the ruler to find its midpoint M
    • C. side = CN: with centre C and radius CN, mark off round the circle five times and join the points in turn
    • D. ∠AOC = 90°: draw a circle, centre O and radius 3 cm, and with a set square two diameters AB and CD at right angles

    Write the letters in the right order

    Answer:

  7. 7.Draw a regular pentagon in a circle of radius 8 cm with compasses, a ruler and a set square: put the steps in order.

    • A. ∠AOC = 90°: draw a circle, centre O and radius 8 cm, and with a set square two diameters AB and CD at right angles
    • B. OM = MB: measure OB with the ruler to find its midpoint M
    • C. side = CN: with centre C and radius CN, mark off round the circle five times and join the points in turn
    • D. MN = MC: with centre M and radius MC, draw an arc cutting OA at N

    Write the letters in the right order

    Answer:

  8. 8.The regular pentagon ABCDE has diagonals AC and AD. ∠BAC = °, ∠CAD = °, ∠DAE = °.

    Fill in each blank

    Answer:

  9. 9.ABCDE is a regular pentagon and AC a diagonal. ∠ACD =

    Answer:

Part B: Choose, or true or false (2 marks each)

  1. 10.Each angle of a regular pentagon is 60°.

    True or false?

    Answer:

  2. 11.Each angle of a regular pentagon is 72°.

    True or false?

    Answer:

  3. 12.Each angle of a regular pentagon is 108°.

    True or false?

    Answer:

Part C: Word problems (4 marks each)

  1. 13.In a regular pentagon ABCDE, the diagonal AC splits ∠BCD into ∠BCA = 40° and ∠ACD = 68°.

    True or false?

    Working:

    Answer:

  2. 14.A regular pentagon tile ABCDE is cut along the diagonal AD. ∠ADC =

    Working:

    Answer:

  3. 15.A coaster is a regular pentagon with sides of 28 cm. Each interior angle = °, each exterior angle = °, perimeter = cm.

    Fill in each blank

    Working:

    Answer:

  4. 16.ABCDE is a regular pentagon. Find ∠CAD: put the steps in order.

    • A. angle sum = (5 − 2) × 180° = 540°
    • B. ∠BAC = ∠DAE = (180° − 108°) ÷ 2 = 36° (base ∠s, isos. △)
    • C. ∠BAE = 540° ÷ 5 = 108°
    • D. ∠CAD = 108° − 36° − 36° = 36°

    Write the letters in the right order

    Working:

    Answer:

  5. 17.In a regular pentagon ABCDE, the diagonal AC splits ∠BCD into ∠BCA = 30° and ∠ACD = 78°.

    True or false?

    Working:

    Answer:

  6. 18.A coaster is a regular pentagon with sides of 13 cm. Each interior angle = °, each exterior angle = °, perimeter = cm.

    Fill in each blank

    Working:

    Answer:

Show the answers and working

Drawing a regular pentagon worksheet: Answers and working

Part A: Working out and filling in

  1. 1. 72°

    The five angles at the centre are equal and make a full turn: x = 360° ÷ 5 = 72°.

  2. 2. 108°

    The angles add to (5 − 2) × 180° = 540°, all five equal: x = 540° ÷ 5 = 108°.

  3. 3. 36°

    AB = BC, so △ABC is isosceles with ∠ABC = 108°: ∠BAC = (180° − 108°) ÷ 2 = 36° (base ∠s, isos. △).

  4. 4. B → C → A → D

    The diameters at right angles come first, so that OB's midpoint M can be found; M gives the arc and N; CN is the side, marked off and joined last.

  5. 5. C → A → B → D

    The diameters at right angles come first, so that OB's midpoint M can be found; M gives the arc and N; CN is the side, marked off and joined last.

  6. 6. D → B → A → C

    The diameters at right angles come first, so that OB's midpoint M can be found; M gives the arc and N; CN is the side, marked off and joined last.

  7. 7. A → B → D → C

    The diameters at right angles come first, so that OB's midpoint M can be found; M gives the arc and N; CN is the side, marked off and joined last.

  8. 8. 36, 36, 36

    ∠BAC = ∠DAE = (180° − 108°) ÷ 2 = 36° (base ∠s, isos. △), so ∠CAD = 108° − 36° − 36° = 36°: the diagonals cut 108° into three equal parts.

  9. 9. 72°

    ∠BCA = (180° − 108°) ÷ 2 = 36° (base ∠s, isos. △), so ∠ACD = ∠BCD − ∠BCA = 108° − 36° = 72°.

Part B: Choose, or true or false

  1. 10. False

    The angles add to (5 − 2) × 180° = 540°, each 540° ÷ 5 = 108°; 72° is its angle at the centre and its exterior angle.

  2. 11. False

    The angles add to (5 − 2) × 180° = 540°, each 540° ÷ 5 = 108°; 72° is its angle at the centre and its exterior angle.

  3. 12. True

    The angles add to (5 − 2) × 180° = 540°, each 540° ÷ 5 = 108°; 72° is its angle at the centre and its exterior angle.

Part C: Word problems

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

  1. 13. False

    BA = BC and ∠B = 108°: ∠BCA = (180° − 108°) ÷ 2 = 36° (base ∠s, isos. △), ∠ACD = 108° − 36° = 72°.

  2. 14. 72°

    ED = EA and ∠E = 108°, so ∠EDA = (180° − 108°) ÷ 2 = 36° (base ∠s, isos. △); ∠ADC = 108° − 36° = 72°.

  3. 15. 108, 72, 140

    Interior = 540° ÷ 5 = 108°, exterior = 360° ÷ 5 = 72°, perimeter = 5 × 28 = 140 cm.

  4. 16. A → C → B → D

    Angle sum → each angle 108° → the base angles of the isosceles triangles either side, 36° → ∠CAD = 36°.

  5. 17. False

    BA = BC and ∠B = 108°: ∠BCA = (180° − 108°) ÷ 2 = 36° (base ∠s, isos. △), ∠ACD = 108° − 36° = 72°.

  6. 18. 108, 72, 65

    Interior = 540° ÷ 5 = 108°, exterior = 360° ÷ 5 = 72°, perimeter = 5 × 13 = 65 cm.

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