Regular Polygons
Introduction

As you can see in the Looking at Polygons unit, regular polygons have equal-length sides and interior angles of equal size.



This regularity of form makes regular polygons useful in a variety of situations, as illustrated above.

Exterior angles
Like any
convex
A polygon is convex if all of its interior angles are less than 180.
convex
polygon
A polygon is a closed, planar figure bounded by straight line segments.
polygon
, the exterior angles of a
regular polygon
A regular polygon has sides of equal length and interior angles of the same size.
regular polygon
sum to 360. Unlike other polygons, however, the exterior angles in a regular polygon are equal. Since a regular n-sided polygon has n exterior angles, each
angle
An angle is a measure of turning. Angles are measured in degrees. The symbol for an angle is .
angle
measures .

Figure 1.   Exterior angles in regular polygons.
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What size is the
exterior angle
When the side of a convex polygon is produced (lengthened), the exterior angle is the angle between this line and an adjacent side.
exterior angle
in a regular 10-sided
decagon
A decagon is a ten-sided polygon.
decagon
?
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What size is the exterior angle in a regular 30-sided polygon?
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Interior angles
All interior angles in a regular polygon are equal. Any
interior angle
An interior angle is the angle between adjacent sides at a vertex of a polygon.
interior angle
added to its adjacent exterior angle is 180, since together they make a
straight line
A straight line is a set of points related by an equation of the form y = ax + c. It has length and position, but no breadth and is therefore one-dimensional.
straight line
.

Figure 2.   Angles at a vertex of a regular polygon.
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We also know that each exterior angle of an n-sided polygon is equal to . So:

    



This
equation
An equation is a mathematical statement that two expressions have equal value. The expressions are linked with the 'equals' symbol (=).
equation
can be used to answer questions like the following:

If the interior angles of a regular polygon are 150, how many sides does it have?

     It has 12 sides.



What size is the interior angle in a regular
nonagon
A nonagon is a nine-sided polygon.
nonagon
?
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How many sides does a regular polygon with interior angles of 175 have?
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Circumscribing regular polygons
Every regular polygon can 'fit' within a circle so that each of its vertices touches the circle edge.
Figure 3.   Circumscribed regular polygons.
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To see why this works, try the following exercise.


Click on the figure below to interact with the model.

 Figure 4.  Making a polygon from isosceles triangles.

  • Click on the Paste new triangle button to paste a copy of the triangle onto the model (it will appear over the first triangle).

  • Click on the new triangle to select it.

  • Use the handle that appears to rotate the triangle until it is next to, but not overlapping, the previous triangle.

  • Now repeat this pasting and rotating process, until there are eight triangles in the model.

What type of polygon have you created?
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Do you notice anything about the eight lengths radiating from
point
  • A point has no properties except position. It is an object with zero dimensions.
  • Points in the x-y plane can be specified using x and y coordinates.
point
C? Because the eight triangles are isosceles and congruent, each of these lengths is the same, 3 cm.

Complete the sentence below with reference to the polygon you have constructed.

  • A circle of radius cm and centre will circumscribe the polygon.
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The fact that any regular n-sided polygon can be divided into n congruent isosceles triangles means that the angle formed at the centre within each
triangle
A triangle is a three-sided polygon.
triangle
will always be:

Figure 5.   An n-sided polygon.
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What is the angle, d, in the regular
dodecagon
A dodecagon is a 12-sided polygon.
dodecagon
in Fig.6 below?
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Figure 6.   A regular dodecagon.
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Constructing regular polygons
The fact that all regular polygons can be circumscribed makes constructing them very easy. We don't need to construct n separate triangles to form an n-sided polygon, we need only do the following:

  1. Choose the number of sides your regular polygon is to have.

  2. Work out the angle, a, formed in the centre of each congruent triangle.

  3. Using your compass draw a circle, centre O.

  4. Draw a line, using a ruler, from O to any point on the circumference of the circle.

  5. Using a protractor and a ruler, draw a line, at an angle a to the first line, from O to the circumference of the circle.

  6. Set your compasses to the distance between the two points at which the first and second lines meet the circle. Starting at one of these points, use your compass to mark off points at this same distance around the circle. These points will become the vertices of your polygon.

  7. Using a straight edge or ruler, join each point to the next point on the circumference of the circle until your polygon is complete.



Enter a value between 3 and 20 for n in Fig.7 below and advance through the stages in the construction process to see the steps in drawing a regular polygon.

Figure 7.   Stages in the construction of a regular polygon.
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A
hexagon
A hexagon is a six-sided polygon.
hexagon
can be formed by six congruent triangles. What kind of triangles are they?
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If you want to construct a hexagon with side length 3 cm, what
radius
A straight line segment joining the centre of a circle with a point on its circumference (or the length of this line).
radius
circle must you draw?
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So constructing a hexagon is extremely easy. No calculations need to be made: if you want to construct a hexagon with side length d cm, you:

  • Draw a circle with radius d cm.

  • Keep the compass set as it is and mark off 6 points on the circle, each one d cm from the last.

  • Join each point to the next one on the circle with a straight line.



Here's a diagram of how your page should look if you draw a hexagon the simple way.

Figure 8.   A constructed hexagon.
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Summary

All exterior angles in a regular polygon are equal. Each is equal to , where n is the number of sides.

All interior angles in a regular polygon are equal. Each is equal to , where n is the number of sides.

Every regular polygon has a circumscribing circle.

Exercises
1. Work out the exterior angles of the shapes in Fig.9 below.

  • Shape Exterior angle
    1
    2
    3 (to 2 s.f.)
    4
    5 (to 2 s.f.)

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Figure 9.   Regular polygons.
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2. What are the interior angles of the shapes in Fig.9 above?

  • Shape Interior angle
    1
    2
    3 (to 3 s.f.)
    4
    5 (to 3 s.f.)

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3. What is a 3-sided regular polygon more often called?

  • It is commonly referred to as a(n)
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4. What is a 4-sided regular polygon more often called?

  • It is commonly referred to as a(n)
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5. What is the sum of the exterior angles for the following polygons?

  • Shape Sum of exterior angles
    Regular pentagon
    Regular octagon
    Regular dodecagon

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6. Which of these formulae can be used to find the exterior angle of a regular n-sided polygon?
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7. Which of these formulae can be used to find the sum of the interior angles of a regular n-sided polygon?
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8. Construct a regular
octagon
An octagon is an eight-sided polygon.
octagon
by inscribing it in a circle of radius 4 cm. Label its vertices A to H in order anticlockwise. Now draw in lines AB, BD, DG, and GA. What
quadrilateral
A quadrilateral is a polygon with four sides.
quadrilateral
have you inscribed in the circle?

  • It is a(n)
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Well done!
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