Adding It All Up
Introduction
Take a look at this activity. Can you name all the polygons? Can you name ones that are not included in the activity? Click here to learn how to count in Greek!
By the end of today's lesson you will hopefully be able to find a rule for interior and exterior angles of polygons.
Investigation
Part 1
Using Geogebra, draw an irregular pentagon (a 5-sided polygon).
Work out the size of the 5 angles, and then add them together, what is the total?
Now try a different pentagon. What is the total this time?
And now try a third pentagon. What is the total this time?
What do you notice about the answer?
Is this always true? Can you find a pentagon for which this isn’t true?
Save the file with the name “5 sides” in a folder called “Adding It All Up”.
Record your results in the table shown on the right (Table 1).
Now repeat the process for a hexagon (6-sides).
Record the results in Table 1.
Repeat the process for other shapes until you have completed Table 1. To speed it up, you wish to use this to investigate 3 to 6 sided polygons.
What do you notice about the results? You might want to check your answers with other students on your table.
What is the rule that connects the number of sides that a shape has with the total of the angles on this inside of the shape?
Can you explain how the why the rule works? Take a look here.
What would the angles on the inside on a 9-sided shape add to? What about a 10-sided shape? What about a 20-sided shape?
Part 2
For regular shapes, all of the internal angles (a.k.a interior angles) must all be the same.
Use this fact to complete the table on the right (Table 2) for regular shapes.
What is the rule for calculating the internal angle of a regular polygon if you know how many sides it has?
What is the interior angle of a regular hendecagon (11 sides)?
What is the interior angle of a regular a dodecagon (12 sides)?
What it the interior angle of a regular icosagon (20 sides)?
A regular polygon has an interior angle of 170°. How many sides does it have?
Part 3
A lot of the shapes have names that end in “-gon”. Why is that? Where does the word come from?
What do the words in front of the “gon” mean, and where do they come from?
Using these rules, what should we call triangles and quadrilaterals?
Part 4
Take a look at this animation.
What is a + b + c + d + e, the total angles turned by the car?
a, b, c, d, e are called the exterior angles of the convex pentagon. What is the relationship between an interior and an exterior angle?
In general, what do you think the sum of all exterior angles of a convex polygon is? Take a look at this to check that your prediction works.
How would you modify the result if the pentagon is NOT convex?
Further Practice
Some of the essential skills introduced in this lesson are interior and exterior angles of triangles, quadrilaterals and polygons. Practise these relevant skills on DrFrostMaths, CorbettMaths, MyiMaths and Eedi.
Transum
Angle Theorems - this is not a practice exercise, it simply lists out all the theorems you need to know.
Angle Points - try this self-marked exercise. There are altogether 4 levels. This is a practice of contents you learnt in year 7.
Angles in a Triangle - try this self-marked exercise. There are altogether 2 levels. This is a practice of contents you learnt in year 7.
Angles with Parallel Lines - try this self-marked exercise. There are altogether 3 levels. This is a practice of contents you learnt in year 7.
Angles in Polygons - try this self-marked exercise. There are altogether 3 levels.
Angles Mixed - try this self-marked exercise. There are altogether 5 levels.
For more goodies on angles on transum, click on the hyperlink.
For some textbook practice, see below:
SMP Book 8C - Chapter 21.pdfExtension
Surrounding a point
In this activity, you will be looking at which combination of convex regular polygons will fit around a point with no overlap and no gaps as shown on in the pictures on the right around the big black dots.
Check that in the two pictures, the polygons fit around a point with no overlap and no gaps. Hint: think about what angles around a point add up to.
Is it possible to use only two polygons? Is it possible to use three different polygons? What about four different polygons, once each?
Conjecture: it’s impossible to have an arrangement with a pentagon in because a pentagon’s angle ends in an ‘8’. Is this true?
Take a look at the following combinations of regular polygons, which of them fit around a point with no overlap and no gaps?
3 hexagons
2 octagons + 1 square
1 octagon + 1 hexagon + 1 square
2 nonagons + 1 square
2 dodecagons + 1 equilateral triangle
1 decagon + 2 pentagons
1 octagon + 1 hexagon + 1 pentagon
1 20-sided polygon + 1 pentagon + 1 square
1 15-sied polygon + 1 decagon + 1 equilateral triangle
1 20-sided polygon + 1 nonagon + 1 equilateral triangle
What other combinations of regular polygons can you find that completely surround a point?
Either use the polygon sheets (PRINT ME) so that you have several copies of each page, and make each page a different colour, i.e. triangles are yellow, squares and blue.
Or use this interactive version - you can either fill a blank piece of plain paper with a single pattern that tessellates, or use a combination of tessellating patterns, or with as many examples as they can of arrangements that surround a point.
What is the smallest number of polygons needed? Why?
What is the biggest number of polygons you can use? Why?
How many combinations can you find that don’t use the same polygon twice?
If you are interested take a look at Euclidean tiling.
nRich
Here is a collection of activities to do with "angles and polygons"
Don Steward
Finally, have a look at the twitter post on the right. What is your opinion? Can you justify your opinion?
