In this section we will be working EXCLUSIVELY with REGULAR polygons!

We will begin with a regular triangle which we call an ______________triangle.

Let's state everything we know about a regular triangle!

Sum of interior angles is 180 degrees.

Sum of exterior angles is 360 degrees.

Each interior angle is 180/3=60 degrees.

Each exterior angle is 360/3=120 degrees.

Here is a picture:

So everything we stated is confirmed!

In class investigation (with partner): 5 minutes

1. ) How do we find the perimeter of a regular triangle?

2.) How do we find the area of a regular triangle?

I will give you an example to practice with!

Now that you have had enough time to experiment, let us find the perimeter.

The Perimeter of any polygon means that we need to

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So the perimeter of this regular triangle is ____cm.

Can we do the same method with a triangle that is not regular?

What about the area? We know that one formula for area of a triangle is (1/2)bh. Let's use that formula to find the area.

1.) The base b is 8 cm.

2.) We do not know the height. We need to draw the altitude. Since the triangle is equilateral, the base is spilt into two equal parts when the altitude intersects it. A right angle is also formed.

3.) How do we find the altitude? YES! Pythagorean Theorem. Let . Solving this for x we get that

4.) Finishing, Area = =

Is there another method to do this?

Assignment #3:

1.) Research the internet or look in the library to find 2 other formulas that will find the area of a regular triangle! (Some may find area of any triangle and that is fine)!

2.) Create a problem similar to the one just completed and find the perimeter and area using one both of the new formulas you found. Also, check your answer by using Area = (1/2)bh.

We also want to find the area of any regular polygon with sides of 10 cm. Let's start with a hexagon and then we will come to a generalization!

Above is a regular hexagon inscribed in a circle. (Remember that inscribed means inside with all vertices touching the circle.)

1.) Break the hexagon into triangles.

What can we say about these triangles?

Notice that there is one orange triangle. That is the triangle we are going to focus on because we know how to find the area of a triangle.

2.) Let's use the method from above to find the area of the triangle. The only question is how do we find the angles of the orange triangle?

3.) Well, we know that all of the interior angles add up to 720 in a hexagon. so each one is 720/6=120. Also, each one is being cut in half, so the angles of the triangle are 60-60-60. EQUILATERAL!!!!

So here is the picture we are working with.

4.) We can find the area of each triangle and then multiply by 6 since there are 6 of them.

So the area of the orange triangle will be A = 1/2bh=1/2(10)(h).

5.) We can use special right triangle rules to find the height. The height will be .

6.) Hence the area will be 1/2(10)()=.

7.) This is only the area of the orange triangle! So, 6()=.

Assignment #4:

1.) Find the area and perimeter of a regular pentagon with side lengths of 6 cm.

2.) Find the perimeter and area of a regular quadrilateral with side lengths of 4 cm.

3.) Come up with a unique generalization for the area of any regular polygon.

Go straight to section 3

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