Area of a circle from what we already know

Author:Alfonso Garcia, Malin Christersson

An interesting surprise

In this section we will explore how to determine the area of a circle by dividing it into sections and then rearranging these sections to form a shape with an easily recognizable area. Follow the instructions and have fun!

1. Slider 1

The applet below shows a circle divided into 4 sections. As we move slider 1 the number of sections increases by multiples of 4. Move slider 1 to 3 and notice that there are now 12 sections in the circle. We will explore what happens when we play with the other two sliders in the sections below.

2. Slider 2

Set slider 1 to 5 (The circle will have 20 sections). Move slider 2 all the way to the left and describe what happens. You can change the value of slider 1 to greater numbers to help in your description.

3. Slider 3

Now move slider 3 all the way down slowly and describe what happens once it gets to the bottom. Give as much detail as possible.

4. What shape do you see?

As slider 3 moved down with a large number of sections for slider 1, what shape is created by rearranging the sections?

Tick all that apply

A
A triangle
B
Many triangles
C
A parallelogram

Check my answer (3)

5. Now, what is the area?

If you answered parallelogram in the previous question, you are correct! Write the formula for the area of a parallelogram.

6. Connecting to the circle part 1

The area of a parallelogram is The height of the parallelogram is the same as...

Tick all that apply

A
the radius of the circle
B
the diameter of the circle
C
the circumference of the circle

Check my answer (3)

7. Connecting to the circle part 2 - the base

The area of a parallelogram is The applet shows the base = Which of the following best describes why this is true?

Tick all that apply

A
Because when the two sections slide to form the parallelogram, their total length is half of the circumference , and since , we have
B
The applet is incorrect, the correct base should be
C
The sections form a rectangle, not a parallelogram
D
Because the base of the parallelogram is the same as the circumference of the circle

Check my answer (3)

8. A little algebra

Simplify the right side of this equation. This is the area of the parallelogram: Comment on your simplified equation.

Area of a circle from what we already know

An interesting surprise

1. Slider 1

2. Slider 2

3. Slider 3

4. What shape do you see?

5. Now, what is the area?

6. Connecting to the circle part 1

7. Connecting to the circle part 2 - the base

8. A little algebra

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