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Regular Polygons

Show your work for each problem.

  1. 1.
    Which of these shapes is a regular polygon?
    1. a)A right triangle
    2. b)A rectangle 4 cm by 7 cm
    3. c)A square
    4. d)A rhombus with angles of 60°60° and 120°120°

    Answer: A square

    Only the square has 4 equal sides and 4 equal angles of 90°90°. The rectangle has unequal sides, the rhombus has unequal angles, and a right triangle cannot have three equal angles.

  2. 2.
    What is each exterior angle of a regular decagon (10 sides), in degrees?

    Answer: 36

    Exterior angle =360°10=36°= \frac{360°}{10} = 36°.

  3. 3.
    Find each interior angle of a regular pentagon.

    Answer: 108

    1. What is the sum of the interior angles, (5−2)×180°(5-2) \times 180°? 540
    2. Divide the sum by 5. How big is each angle? 108
  4. 4.
    What is each interior angle of a regular hexagon?
    1. a)120°120°
    2. b)108°108°
    3. c)60°60°
    4. d)720°720°

    Answer: 120°120°

    (6−2)×180°6=720°6=120°\frac{(6-2) \times 180°}{6} = \frac{720°}{6} = 120°. 720°720° is the sum of all six angles, 60°60° is the exterior and central angle.

  5. 5.
    A regular octagon has sides of 7 cm. What is its perimeter in cm?

    Answer: 56

    P=8×7=56P = 8 \times 7 = 56 cm

  6. 6.
    A regular nonagon has 9 sides. Find its exterior angle and then its interior angle.

    Answer: 140

    1. What is the exterior angle, 360°9\frac{360°}{9}? 40
    2. What is the interior angle, 180°−40°180° - 40°? 140
  7. 7.
    Which regular polygon can tile a floor on its own, with no gaps and no overlaps?
    1. a)Regular hexagon
    2. b)Regular pentagon
    3. c)Regular decagon
    4. d)Regular octagon

    Answer: Regular hexagon

    The hexagon's interior angle is 120°120° and 3×120°=360°3 \times 120° = 360°. The pentagon (108°108°), octagon (135°135°) and decagon (144°144°) have angles that do not divide 360°360°.

  8. 8.
    What is each interior angle of a regular 12-sided polygon (dodecagon), in degrees?

    Answer: 150

    (12−2)×180°12=1800°12=150°\frac{(12-2) \times 180°}{12} = \frac{1800°}{12} = 150° (or 180°−360°12=180°−30°=150°180° - \frac{360°}{12} = 180° - 30° = 150°).

  9. 9.
    A regular hexagon with sides of 5 cm is cut into 6 triangles from its center. Find the angles of one triangle and the radius of the hexagon.

    Answer: 5

    1. What is the central angle of a regular hexagon, in degrees? 60
    2. Each triangle is isosceles. What is each base angle, 180°−60°2\frac{180° - 60°}{2}? 60
    3. All angles are 60°60°, so the triangles are equilateral. How long is the radius (center to vertex), in cm? 5
  10. 10.
    The central angle of a regular polygon is 30°30°. How many sides does it have?
    1. a)1212
    2. b)99
    3. c)66
    4. d)1010

    Answer: 1212

    n=360°30°=12n = \frac{360°}{30°} = 12: a regular dodecagon.

  11. 11.
    Each interior angle of a regular polygon is 140°140°. How many sides does it have?

    Answer: 9

    Exterior angle 180°−140°=40°180° - 140° = 40°, so n=360°40°=9n = \frac{360°}{40°} = 9.

  12. 12.
    Each interior angle of a regular polygon is 156°156°. How many sides does it have?

    Answer: 15

    1. What is the exterior angle, 180°−156°180° - 156°? 24
    2. How many sides: n=360°24°n = \frac{360°}{24°}? 15
  13. 13.
    A regular octagon has sides of 10 cm and an apothem of 12.07 cm. Find its area.

    Answer: 482.8

    1. What is the perimeter in cm? 80
    2. What is the area in cm², 12⋅80⋅12.07\frac{1}{2} \cdot 80 \cdot 12.07? 482.8
  14. 14.
    A regular hexagon has sides of 4 cm. How long is its apothem?
    1. a)23≈3.462\sqrt{3} \approx 3.46 cm
    2. b)22 cm
    3. c)88 cm
    4. d)44 cm

    Answer: 23≈3.462\sqrt{3} \approx 3.46 cm

    Each triangle is equilateral with side 4 cm. Its height splits the base into 2 cm and 2 cm: a=42−22=12=23≈3.46a = \sqrt{4^2 - 2^2} = \sqrt{12} = 2\sqrt{3} \approx 3.46 cm. 4 cm is the radius, not the apothem.

  15. 15.
    A regular pentagon has a perimeter of 60 cm and an apothem of 8.26 cm. What is its area in cm²?

    Answer: 247.8

    A=12⋅60⋅8.26=30⋅8.26=247.8A = \frac{1}{2} \cdot 60 \cdot 8.26 = 30 \cdot 8.26 = 247.8 cm²

  16. 16.
    A regular hexagon has sides of 8 cm. Find its area by splitting it into 6 equilateral triangles.

    Answer: 166.32

    1. The apothem is the height of one triangle: 82−42\sqrt{8^2 - 4^2}. Round to 2 decimal places (cm). 6.93
    2. Area of one triangle: 12⋅8⋅6.93\frac{1}{2} \cdot 8 \cdot 6.93 (cm²)? 27.72
    3. Area of the hexagon: 6 triangles of 27.72 cm² (cm²)? 166.32
  17. 17.
    Can regular octagons tile a floor on their own? Place whole octagons around one corner point and find the gap that is left, in degrees.

    Answer: 90

    1. What is the interior angle of a regular octagon, in degrees? 135
    2. How many whole octagons fit around one point without overlapping? 2
    3. How many degrees are left as a gap, 360°−2×135°360° - 2 \times 135°? 90
  18. 18.
    In a regular polygon, each interior angle is 4 times as large as each exterior angle. How many sides does it have?

    Answer: 10

    1. Call the exterior angle ee. Then 4e+e=180°4e + e = 180°. What is ee in degrees? 36
    2. What is the interior angle, 4×36°4 \times 36°? 144
    3. How many sides: n=360°36°n = \frac{360°}{36°}? 10
Interactive lesson with step-by-step solutions and more practice (free): mathorio.com/en/learn/geometry/polygons/regular-polygons
Today's challengeOne problem, three tries