People like to say that one full turn corresponds to 360 degrees of turning and use a \(\circ\) superscript  to denote a count of degrees.

Question: Why that number? Who chose the number 360 for the count of degrees in a circle? Would you think to choose that number? Let’s talk about this in the next (optional) lesson.

Thus half a turn corresponds to \(\dfrac{1}{2}\cdot 360^{\circ}=180^{\circ}\) of turning, quarter of a turn to \(\dfrac{360^{\circ}}{4}=90^{\circ}\) of turning, two full turns to \(2 \times 360^{\circ}=720^{\circ}\) of turning, and so on.

Question: To how many degrees do the following fractions of turn correspond?

a) \(\dfrac{1}{6}\) of a turn            b)  \(\dfrac{3}{4}\) of a turn              c) one-and-a-half turns

To what fraction of a turn do the following counts of degrees correspond?

d) \(45^{\circ}\)           e) \(20^{\circ}\)          f) \(1080^{\circ}\)          g) \(0.01^{\circ}\)

(See the teacher guide here for solutions.)

People throughout the ages have noticed that there seems to be something special about the three angles inside a triangle—any triangle. It appears that

The three interior angles of a triangle have measures sure to add to half a turn (\(180^{\circ}\)).

And pencil pushing shows this. Try this next activity!

STEP 1: Draw a large triangle on a piece of paper, one large enough to contain a small pencil. (Or draw a large triangle on a chalkboard, or in the sand.)

Place a pencil with one end in the corner of the triangle, aligned with the side of a triangle. Take note of the direction the pencil is initially pointing.

STEP 2: Apply to the pencil the amount of turning of the first angle.

STEP 3: Slide the pencil along the side of the triangle on which it sits. This action applies no turning to the pencil.

STEP 4: Apply to the pencil the amount of turning of the second angle.

 STEP 5: Slide the pencil along the side of the triangle on which it sits. This action again applies no turning to the pencil.

STEP 6: Apply to the pencil the amount of turning of the third angle.

 STEP 7: Slide the pencil back to start along the side of the triangle on which is sits.

We see that the pencil is now pointing in the opposite direction to which it started!

The total effect of applying the amount of turning given by the three angles in a triangle is that of a half turn. So indeed,

The three interior angles of a triangle have measures that sum to \(180^{\circ}\).

Can you see in your mind’s eye that for any triangle you happen to draw on the page the same half turn of a pencil would occur? Can you see that the same phenomenon would occur for a pencil moving inside a triangle the size of the solar system? Can you see that it would occur too for a pencil inside a triangle the size of an atom (a sub-atomic sized pencil, of course)? It seems this pencil experiment demonstrates an idea that holds for all triangles.

Some might worry that the turning in step 4 is inconsistent with the other two actions of turning: here we are turning about the lead-tip of the pencil rather than the eraser end of the pencil.

To be consistent, in step 4 we could turn the pencil about the same eraser end too by sliding the pencil just past the vertex of the triangle and make use of the congruent vertical angles we noted in the last lesson.

Turning on the tip or turning on the eraser applies the same amount of turning.

Comment: This animation from Magic Pi Animations is fun.

Please join the conversation on Facebook and Twitter and kindly share this page using the buttons below.

Facebook

Twitter

Resources