Mathematics notes, chapter 4

Polygon

Textbook Mathematics Form 2 (DLP), pages 54–73

What this chapter covers

Describe the geometric properties of regular polygons (equal sides, equal interior angles, n axes of symmetry) and tell regular from irregular polygons (4.1.1)
Construct regular polygons (equilateral triangle, square, hexagon and others) with a ruler and compasses or by dividing the angle at the centre, and explain the steps (4.1.2)
Derive and use the formula (n − 2) × 180° for the sum of the interior angles of a polygon (4.2.1)
Make and verify the conjecture that the exterior angles of a polygon add up to 360° (4.2.2)
Find the interior angle, the exterior angle and the number of sides of a regular polygon, and unknown angles in other polygons (4.2.3)
Solve problems involving polygons, including combinations of polygons (4.2.4)

Polygons are everywhere in buildings and in art: the floating mosque at Tanjung Bungah in Penang uses a geometric pattern of polygons. In Form 1 you met triangles and quadrilaterals. In this chapter you look at regular polygons with any number of sides, how to construct them, and the rules for their interior and exterior angles. The word "polygon" comes from poly (many) and gon (angle).

English	Bahasa Melayu
regular polygon, irregular polygon	poligon sekata, poligon tak sekata
axis of symmetry	paksi simetri
side, vertex	sisi, bucu
interior angle, exterior angle	sudut pedalaman, sudut peluaran
supplementary angle	sudut penggenap
conjecture	konjektur

4.1Regular polygons

Geometric properties of regular polygons

A regular polygon has all sides equal in length and all interior angles equal in size. A polygon that is not regular is an irregular polygon.

Regular pentagon with equal sides and equal angles; rhombus with equal sides but unequal angles; rectangle with equal angles but unequal sides; irregular pentagon — Both conditions are needed for a regular polygon

Equal sides alone are not enough (a rhombus), and equal angles alone are not enough (a rectangle). A regular polygon needs both. A square is the regular quadrilateral.

The textbook (p. 57) also sorts polygons by shape. A convex polygon has no interior angle more than 180°. A concave polygon has at least one interior angle more than 180°.

A convex pentagon and a concave arrowhead shape with one reflex interior angle — Convex and concave polygons (textbook p. 57)

Based on the textbook, Mathematics Form 2 (DLP), p. 57.

Axes of symmetry

An axis of symmetry is a fold line: the two halves match exactly. The activity on p. 58 shows the rule for regular polygons.

A regular polygon with n sides has n axes of symmetry. For an irregular polygon you must find the axes by folding or by looking: a rectangle has 2, a kite 1 and a scalene triangle none.

Regular polygon	Sides	Vertices	Axes of symmetry
equilateral triangle	3	3	3
square	4	4	4
regular pentagon	5	5	5
regular hexagon	6	6	6
regular octagon	8	8	8

State the number of axes of symmetry of a) a regular decagon, b) a rhombus that is not a square, c) a regular polygon with 15 sides. (Based on textbook Self Practice 4.1, p. 61)

Show answerHide answer

a) 10 (a decagon has 10 sides)
b) 2 (a rhombus is irregular: its axes are its two diagonals)
c) 15

Constructing regular polygons

The textbook (pp. 59-61) builds regular polygons with a ruler and compasses. The idea is always that the compasses keep the same radius, so the sides come out equal.

Method: regular hexagon with side 3.5 cm (textbook p. 60)

Draw a circle of radius 3.5 cm and mark a point A on it.
With the compasses still at 3.5 cm and the point on A, mark B on the circle.
Move to B and mark C, then D, E and F in the same way.
Join AB, BC, CD, DE, EF and FA.

A circle with a regular hexagon ABCDEF inside it, centre O joined to every vertex, angle AOB = 60 degrees

Why it works: the radius of the circle equals the side, so each triangle such as OAB has three equal sides. It is equilateral, so ∠AOB = 60°, and 6 × 60° = 360° fills the centre exactly.

Other constructions in the textbook: an equilateral triangle of side 5 cm (two arcs of radius 5 cm from the ends of the base meet at the third vertex), and a square of side 4 cm (a perpendicular at A, an arc of 4 cm to mark D, then two arcs of 4 cm from B and D meet at C).

Dividing the angle at the centre: the angle at the centre between neighbouring vertices is 360° ÷ n. For a regular pentagon it is 360° ÷ 5 = 72°, so mark five points 72° apart on a circle (with a protractor) and join them.

A regular polygon drawn with a protractor and ruler is less accurate than one built with compasses or dynamic geometry software, because small errors add up. Always keep the compass radius fixed while stepping round.

a) How large is the angle at the centre between two neighbouring vertices of a regular octagon? b) A regular nonagon is to be drawn by dividing the centre of a circle equally. How many degrees apart are the vertices? (Textbook Self Practice 4.1 Q5, p. 62)

Show answerHide answer

a) 360° ÷ 8 = 45°
b) 360° ÷ 9 = 40°

4.2Interior angles and exterior angles of polygons

Interior and exterior angles

An interior angle is formed inside the polygon by two adjacent sides. An exterior angle is formed when one side is extended; it lies outside the polygon next to the interior angle. They are supplementary: interior angle + exterior angle = 180°.

Pentagon with one vertex showing interior angle a inside, a side extended, and exterior angle b outside; a + b = 180 degrees — a is an interior angle, b is the exterior angle at the same vertex: a + b = 180°

Sum of the interior angles

Choose one vertex and draw diagonals to the other vertices. This cuts an n-sided polygon into (n − 2) triangles. Each triangle has angles totalling 180°.

Pentagon cut into 3 triangles, hexagon into 4 triangles, heptagon into 5 triangles by diagonals from one vertex — Number of triangles = n − 2 (textbook p. 62)

Polygon	Sides n	Triangles	Sum of interior angles
triangle	3	1	1 × 180° = 180°
quadrilateral	4	2	2 × 180° = 360°
pentagon	5	3	3 × 180° = 540°
hexagon	6	4	4 × 180° = 720°
octagon	8	6	6 × 180° = 1 080°
n-sided polygon	n	n − 2	(n − 2) × 180°

Sum of the interior angles of an n-sided polygon = (n − 2) × 180°.

The names of larger polygons (textbook p. 63): 12 dodecagon, 13 tridecagon, 14 tetradecagon, 15 pentadecagon, 16 hexadecagon, 17 heptadecagon, 18 octadecagon, 19 enneadecagon, 20 icosagon.

Example 1 (textbook p. 63)

Find the number of triangles and the sum of the interior angles of a polygon with a) 13 sides, b) 18 sides.

a) Triangles = 13 − 2 = 11. Sum = 11 × 180° = 1 980°
b) Triangles = 18 − 2 = 16. Sum = 16 × 180° = 2 880°

Example 2 (based on textbook p. 64)

Find the value of x in each polygon.

(a) Pentagon with interior angles 110, 125, 95, 140 degrees and x. (b) Quadrilateral with interior angles 100, 75, 110 degrees and x

(a) Sum for a pentagon = (5 − 2) × 180° = 540°
x + 110° + 125° + 95° + 140° = 540°, so x + 470° = 540° and x = 70°
(b) Sum for a quadrilateral = (4 − 2) × 180° = 360°
x + 100° + 75° + 110° = 360°, so x = 360° − 285° = 75°

Subtract 2 before multiplying: (n − 2) × 180°, not n − 2 × 180°. A hexagon has (6 − 2) × 180° = 720°, not 6 × 180° = 1 080°.

a) Find the sum of the interior angles of a heptagon, a 9-sided polygon and a 12-sided polygon. b) Four interior angles of a pentagon are 120°, 100°, 130° and 70°. Find the fifth angle. (Based on textbook Self Practice 4.2, p. 66)

Show answerHide answer

a) Heptagon: 5 × 180° = 900°. 9 sides: 7 × 180° = 1 260°. 12 sides: 10 × 180° = 1 800°.
b) 540° − (120° + 100° + 130° + 70°) = 540° − 420° = 120°

Sum of the exterior angles

Walk once round a polygon: at every vertex you turn through its exterior angle, and after one complete round you have turned a full circle. The textbook confirms this by exploring polygons with different n.

The sum of the exterior angles of any polygon (one at each vertex) is 360°, whatever the number of sides.

A conjecture is a statement that looks right from examples but has not yet been proved (textbook p. 64). The exterior-angle rule is first a conjecture from the dynamic geometry activity, then it is checked with polygons of 3, 4, 5, 6 ... sides.

Example 3 (based on textbook p. 65)

The exterior angles of a pentagon are 85°, 100°, 70°, x and 2x. Find x and the interior angle at the vertex with exterior angle 2x.

Pentagon with a side extended at every vertex, exterior angles labelled 85, 100, 70, x and 2x degrees

Sum of exterior angles = 360°
85° + 100° + 70° + x + 2x = 360°, so 255° + 3x = 360°
3x = 105°, so x = 35° and 2x = 70°
Interior angle = 180° − 70° = 110°

The sum of the exterior angles is 360° for every polygon. Only the sum of the interior angles changes with n.

Regular polygons: finding angles and sides

For a regular polygon with n sides:
Exterior angle = 360°n
Interior angle = (n − 2) × 180°n or 180° − exterior angle
Number of sides n = 360°exterior angle

Example 4 (based on textbook pp. 65-66)

For a regular decagon (10 sides), find a) the exterior angle, b) the interior angle.

a) Exterior angle = 360° ÷ 10 = 36°
b) Interior angle = 180° − 36° = 144°
Check with the sum: (10 − 2) × 180° = 1 440°, and 1 440° ÷ 10 = 144°.

Example 5 (based on textbook p. 66)

Find the number of sides of a regular polygon when an interior angle is a) 156°, b) 140°.

a) Exterior angle = 180° − 156° = 24°, so n = 360° ÷ 24 = 15
b) Exterior angle = 180° − 140° = 40°, so n = 360° ÷ 40 = 9

Example 6

The sum of the interior angles of a polygon is 2 340°. How many sides has it?

(n − 2) × 180° = 2 340°
n − 2 = 2 340 ÷ 180 = 13
n = 15

Example 7

In a regular polygon the interior angle is (4y + 20)° and the exterior angle is (y + 10)°. Find the number of sides.

Interior + exterior = 180°: 4y + 20° + y + 10° = 180°, so 5y = 150° and y = 30°
Exterior angle = 30° + 10° = 40°
n = 360° ÷ 40 = 9 sides (a nonagon). The interior angle is 4(30°) + 20° = 140°.

A regular polygon's number of sides must be a whole number. If 360° ÷ exterior angle is not whole, such a regular polygon does not exist. For example an interior angle of 100° gives an exterior angle of 80° and 360 ÷ 80 = 4.5 sides, so it is impossible.

a) Find the exterior and interior angle of a regular polygon with 20 sides. b) The exterior angle of a regular polygon is 18°. How many sides has it? c) Can a regular polygon have an interior angle of 130°? Explain. (Based on textbook Self Practice 4.2, p. 67)

Show answerHide answer

a) Exterior = 360° ÷ 20 = 18°, interior = 180° − 18° = 162°
b) n = 360° ÷ 18 = 20
c) Exterior angle = 50° and 360 ÷ 50 = 7.2, which is not a whole number, so no.

Solving problems involving polygons

The textbook (p. 66) uses four steps: understand the problem, plan the strategy, carry out the strategy, and conclude (check). Draw the figure, mark what you know, and give a reason for each angle.

Example 8 (based on textbook p. 64)

ABCDE is a regular pentagon. BC and ED are extended to meet at F. Find x = ∠CFD.

Regular pentagon ABCDE with sides BC and ED extended beyond C and D to meet at F above side CD; x is angle CFD

Exterior angle of a regular pentagon = 360° ÷ 5 = 72°
∠FCD = ∠FDC = 72° (exterior angles at C and D)
In triangle CFD: x = 180° − 72° − 72° = 36°

For regular-polygon problems, find the interior or exterior angle first. The exterior angle 360° ÷ n is usually the quickest.

Summary

Idea	Key fact	Example
Regular polygon	All sides equal and all interior angles equal	A rhombus is not regular
Axes of symmetry	Regular n-gon has n axes	Regular hexagon: 6
Construction	Compasses keep the side equal; angle at the centre = 360° ÷ n	Hexagon: 6 triangles of 60°
Interior angle + exterior angle	180° at the same vertex	120° + 60°
Sum of interior angles	(n − 2) × 180°	Octagon: 1 080°
Sum of exterior angles	360° for every polygon	Always 360°
Regular polygon	Exterior = 360° ÷ n; interior = 180° − exterior	n = 9: 40° and 140°
Number of sides	n = 360° ÷ exterior angle	Exterior 24°: n = 15