How would you use a model to determine the approximate formula for the area of a circle with a diameter of 8 cm, and connect the model to the actual formula?

Hello!

There are many different models for the area of a circle. The way I learned it has to do with parallelograms and multiplying.

To start out, let's note that our radius is 4 cm. If we were to cut the circle into equal triangular pieces, we can combine them to make a "parallelogram". We can lay a few of the facing up at the bottom, leaving some spaces of up side down triangles. Then we put our remaining triangles from the circle into the spots making a parallelogram.

To make this sound less complicated, lets say that we took the top of  the circle and "unfolded" it. This would give us the parallelogram. with the top of the parallelogram containing the middle point of the original circle.

To find the area of the circle, we need to find the area of this parallelogram. First we need to find the length of the base. As the circumference of the circle was unfolded to make the bases, we know that half of it made the top base, and the other half made the bottom. Half of the circumference would be half the diameter, which is the radius multiplied by pi. In our case, the radius is 4, so 4 times pi is equal to the base. 4.

Then, we need to find the height. Before I said that the middle of the circle is located at the top of the parallelogram, as the circle "unfolded" around the origin. Let's say that the top of our base is the center of the circle. The distance from the outside to the center is the radius, so the height is 4.

Now, we need to multiply the base and the height. 4·4=A. This means that 16 is equal to the area. The formula for the area of a circle is A=r². No matter what the radius is, you will have to multiply it by itself and pi, as seen in 4·4. 4·4 is 4², where 4=r. This proves the formula for the area of a circle.

I hope this was clear, and understandable! I hope this helps you!

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