Triangle exploration and investigation activities

Introduction:TYpes of triabgles

Activities include ideas related to geometry, patterns, problem solving, reasoning, visual representation, trigonometry, and other mathematical ideas

Trigonometry is the study of triangles and the relations of their sides and angles.

A triangle is a plane closed figure with three straight sides and three angles. There are three kinds based on the length of their sides.

Activity ideas

Triangles Venn diagram Triangle tree

Triangle forensics

Torn triangles

Can a triangle be recovered from it's parts?


Can triangles be made with any length sides?

What lengths of sides will make triangles? Use straws, toothpicks, strips of paper, Cuisenaire rods, or other appropriate objects and make a list of combination of lengths that will make triangles and those which will not.

To start focus your investigation by limiting the length of the sides by focusing on 3 cm lengths and see what whole number lengths can be used with it to make triangles.

Let's start with all the sides 3 cm. Will that make a triangle?


Side 1 Side 2 Side 3 Triangle
A 3 cm 3 cm 3 cm Yes - No

Next. Let's try 3 cm on two sides. What whole number combinations will make a triangle?


Side 1 Side 2 Side 3 Triangle
B 3 cm 3 cm 1 cm Yes - No
C 3 cm 3 cm 2 cm Yes - No
D 3 cm 3 cm 4 cm Yes - No
E 3 cm 3 cm 5 cm Yes - No
F 3 cm 3 cm 6 cm Yes - No
G 3 cm 3 cm 7 cm Yes - No


What did you discover?

If some combinations don't work, why didn't they?



Now let's try 3 cm on one side. What whole number combinations will make a triangle?


Side 1 Side 2 Side 3 Triangle
H 3 cm     Yes - No
I 3 cm     Yes - No
J 3 cm     Yes - No
K 3 cm     Yes - No
L 3 cm     Yes - No
M 3 cm     Yes - No
N 3 cm     Yes - No
O 3 cm     Yes - No



What did you discover?

If some combinations don't work, why didn't they?



What conjecture can be made about sides of equal lengths (equilateral) and triangles? Any length of a side can be used to make equilateral triangles.

What conjecture can be made about sides of triangles with two sides equal (isosceles)? The third side of an isosceles triangle is less than the sum of the two sides.

What conjecture can be made about the relationship of one side of a triangle to the other two sides?


Triangle angles

triangle anglesQuestions:


Use a ruler and draw a fairly large triangle on a sheet of paper. Tear off the corners (angles, vertices) and place them together along a line. Does the sum of the angles match the line.


Use a protractor to measure the angles, add them and see how close it is to 180°.

Making a triangle from three angles

The set up



Changing sides & angles proportionally

What happens to the shape of triangles when their angles and sides are changed?

It seems obvious when an angle or side is changed, the other two sides or angles have to change for the shape to be a triangle. Here are some changes to explore:


Angle manipulated Angle not changed Angle responding Side not changed Side responding Side responding
Start 15° 90°   3 cm    
Double once 30° 90°   3 cm    
Double twice 60° 90°   3 cm    


Write conjectures as if, then statements.

If angle A increases, then ...

If angle A is the largest angle, then side a is ...

If angle A is the smallest angle, the side a is ...


What's my angle

What is the relationship of angles in triangles?

Find the angles for the triangle information in the table.

Side a is opposite angle A, b opposite B, and c opposite C. hint: make a drawing.


Angle A Angle B Angle C Side a Side b Side c
A 50° 60° °      
B 32° 108° °      
C ° ° ° a = b= c b = a= c c = a= b
D 66° ° ° a = b b = a  
E 90° 30° °      
F ° 45° °   b = c c = b


What triangle angles will tessellate in a circle fully?

Select a triangle and one of its angles. Place it in the middle of a piece of paper with the selected angle at the center of the circle. Trace it. Rotate it right or left the number of degrees of the angle, then trace it again. Continue until you get to the beginning. Did it rotate a perfect 360?

Try other triangles and find what kinds of shapes the different triangles make.

Which shapes are used in designs?

What would happen if you switched different triangles?

Try it and find shapes and patterns you like.


Using two angles and a side to find another side

Problem types: measuring the height of building, towers, trees, ...

Information: we know the tree, building or tower should be perpendicular to the Earth (90). We can pace or measure a certain distance from the base of the tree or other object to determine a side. Then use a clinometer to find the angle at that point a line would make drawn from that point to the top of the tree or building. How to make and use a clinometer to measure height.




Dr. Robert Sweetland's notes
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