Polygons

Polygons

• the sides are straight lines

• the shapes are closed

• none of the line segments can cross one another

Multiple Choice

The diagram shows two adjacent sides of a regular polygon. What is the name of the polygon?

• Nonagon

• Decagon

• Hendecagon

• Dodecagon

Multiple Select

Which one is not a polygon?

Multiple Select

Choose all the polygons

Polygons

• the sides are straight lines

• the shapes are closed

• none of the line segments can cross one another

Multiple Choice

Name the polygon. Determine if it appears to be regular or not regular.

• 8 sides, not congruent, not regular octagon

• 8 sides, congruent, regular octagon

Multiple Choice

Name the polygon. Determine if it appears to be regular or not regular.

• quadrilateral; regular

• quadrilateral; not regular

• pentagon;regular

• pentagon; not regular

Multiple Choice

Name the polygon. Determine if it appears to be regular or not regular.

• quadrilateral; regular

• quadrilateral; not regular

• hexagon; regular

• hexagon; not regular

Multiple Choice

Name the polygon. Determine if it appears to be regular or not regular.

• quadrilateral; regular

• quadrilateral; not regular

• hexagon; regular

• hexagon; not regular

Multiple Choice

Name the polygon. Determine if it appears to be regular or not regular.

• quadrilateral; regular

• quadrilateral; not regular

• octagon; regular

• octagon; not regular

Polygons

• Regular : All sides and angles are Congruent

• Irregular : Any polygon that is not Regular.

• Concave : A line that contains a side passes outside the polygon.

• Convex : No line that contains a side passes through the polygon.

Parts of a Polygon

• Vertex : Point where two sides intersect.

• Side : Line Segment between vertices.

• Interior Angle : Angle inside the polygon formed by two sides.

• Exterior Angle : Angle outside the polygon formed by two sides.

• Diagonal : Segment that connect two non-consecutive vertices.

Multiple Choice

A quadrilateral has 4 sides, 4 angles and 4 __________.

• Vertices

• Degrees

• Rays

• Parallels

Multiple Select

Which of the following shapes are not polygons?

Multiple Choice

Which polygon has 14 diagonals?

• Hexagon

• Octagon

• Nonagon

• Heptagon

Multiple Choice

What is a regular polygon?

• A polygon that has 1 set of parallel sides and 1 right angle.

• A polygon that has 2 sets of parallel sides and 2 right angles.

• A polygon that has 2 right angles and 2 sides congruent.

• A polygon in which all sides and all angles are congruent.

Multiple Choice

A nonadecagon is a 19-sided polygon.

• True

• False

Angles in Polygons

The exterior angles of a polygon add up to 360^o

Multiple Choice

The exterior angles of a polygon add up to...

• 360^o

• 180^o

• 200^o

• 90^o

Multiple Choice

What is the size of angle a?

• 72

• 55

• 50

• 53

Multiple Choice

What is the size of the exterior angle of a pentagon?

• 100^o

• 360^o

• 72^o

• 180^o

Multiple Choice

What is the size of the exterior angle of a hexagon (six sides) ?

• 360^o

• 50^o

• 60^o

• 180^o

Multiple Choice

What is the size of one exterior angle of a decagon (10 sides) ?

• 90

• 180

• 360

• 36

Multiple Choice

The exterior angle of a regular polygon is 120^o. How many sides does this polygon have?

• 3 sides

• 4 sides

• 5 sides

• 6 sides

Sum of Interior Angles of a Polygon

Use the formula 180n-360, where n is the number of sides of the polygon

Finding the missing angle

First use the formula 180n-360 to find the sum of interior angles. Then add the other angles and subtract to find the missing angle.

Multiple Choice

Work out the missing angle

• 92

• 88

• 103

• 360

Multiple Choice

Find the value of the missing angle

• 96

• 86

• 120

• 116

Multiple Choice

Find the value of the missing angle

• 83

• 112

• 132

• 118

Multiple Choice

Find the value of the missing angle

• 140

• 156

• 132

• 162

Interior angles of regular polygons

Find the sum of interior angles using the formula 180n-360 as before. Then divide by the number of sides to find the value of one angle.

Multiple Choice

Work out the size of one interior angle of this regular hexagon

• 120

• 720

• 100

• 360

Multiple Choice

Work out the size of one interior angle of this regular heptagon

• 143.56

• 130.5

• 128.57

• 150.54

Multiple Choice

Work out the size of each interior angle of this decagon

• 150

• 130

• 144

• 154

Multiple Choice

What is the perimeter?

• 24 sq cm

• 24 cm

• 35 sq cm

• 35 cm

​If one interior angle of the polygon is 150°, and n is the number of sides, then:

⇒ (n - 2) × 180°/n = 150°
⇒ (n - 2) × 180 = 150n
⇒ 180n - 360 = 150n
⇒ 180n - 150n = 360
⇒ 30n = 360
⇒ n = 360/30 = 12

So it is a 12-sided polygon, or dodecagon

The Interior Angle is 135°, so the Exterior Angle is (180° - 135°) = 45°

The total of all Exterior Angles is 360°, and 360°/45° = 8

So it is an 8-sided polygon, or octagon

Multiple Choice

What is the sum of the interior angles of a regular nonagon?

• 980°

• 1,260°

• 1,296°

• 1,620°

​One interior angle of a polygon with n sides = (n - 2) × 180°/n

In this case n = 9
So one interior angle = 7 × 180°/9 = 7 × 20° = 140°

So the sum of the interior angles of a regular nonagon = 9 × 140° = 1,260°

Multiple Choice

What is the sum of the interior angles of a regular pentadecagon (which has 15 sides)?

• 2,340°

• 1,350°

​One interior angle of a polygon with n sides = (n - 2) × 180°/n
In this case n = 15, so one interior angle = 13 × 180°/15 = 13 × 12° = 156°

So the sum of the interior angles of a regular pentadecagon = 15 × 156° = 2,340°

Multiple Choice

The diagram shows a complex pentagon.However, although it's complex, it is symmetrical (all its sides and angles are equal).It is known as a Pentagram or a 5-pointed star.

What is the size of the angle at each of the points A, C, E, G and I?

• 21.6°

• 36°

​Let's look at triangle ICF.

The Angle IFC is one of the angles of the "inside" regular pentagon BDFHJ. A regular pentagon has internal angles of 108°, so we have one of the angles.

Because of the symmetry of the pentagram, the two other angles must be equal.

And a triangles angles add up to 180°

So each of the other two angles of triangle ICF must be ½(180° - 108°) = 36°

So the angle at each of the points A, C, E, G and I = 36°


Alternative Solution
====================
Let's look at triangle ABJ.
The angle HJB is one of the angles of the "inside" regular pentagon BDFHJ.
A regular pentagon has internal angles of 108°, so angle HJB = 108°.
Angle HJA is a straight line, meaning angle HJA = 180°, so angle AJB = 180° - 108° = 72°.
Because the pentagram is symmetrical, angle ABJ = angle AJB = 72°.
The angles of a triangle add up to 180°, so angle JAB = 180° - 2 × 72° = 36°.
Again, because the pentagram is symmetrical, the angle at each of the points A, C, E, G and I is the same 36°.

Multiple Choice

What is the size of the angle at each vertex of the regular 9-pointed star?

(Hint: an interior angle of a regular nonagon is 140°)

• 20°

• 40°

​Consider triangle ABC and the angles marked x° and y°

Because one interior angle of a regular nonagon is 140° ⇒ x° + 2y° = 140° (1)

Because the angles of triangle ABC add to 180° ⇒ 3x° + 2y° = 180° (2)

Subtract equation (1) from equation (2) ⇒ 2x° = 40° ⇒ x = 20°

(Interestingly it follows that y = 60°, so each of those triangles around the perimeter of the nonagon is equilateral)