Area of a Circle and What you Need to Know About Pi

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Introduction to Pi

Pi.(π) Every student will be introduced to this mysterious creature. Everyone of them has been told that it represents the ratio of the circumference of a circle to the diameter. With that in mind, please understand that the area of a circle is equal to πr2. Simple concept! Let's practice using this formula with the following worksheet, and by the way if it makes no sense, then memorize the formula and the fact that you feel 'dumb' is hidden from all.
The ratio π (p) can be demonstrated and with some ingenuity the concept can become concrete using props and hands on . Using π (p) in the context of explaining the area of a circle is another matter. Most students have no tangible evidence or understanding of the use of π (p) in this equation. Sometimes you need to look at the problem from a different point of view to gain some understanding.

Prerequisite: Definitions for circle, radius, diameter, Area of a square.

 

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Step 1.

Draw a circle with a diameter of 6. Calculate the area of this circle using the formula πr2.

 

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Step 2.

Complete the square in the top left quadrant of your circle, using the radius. In other words, in the top left quarter of your circle, draw two line segments 3 units long so as to complete the square ( diagram ).

 

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Step 3.

  From the above, it is easy to see that if the radius of this circle is 3, the top left corner square would have two sides of a length of 3 and therefore the square would have an area of 9. So what is the area of the segment of circle interior that is in this square ? After visual review, it is simple to see that this represents one quarter of the circle, so it can be represented by πr2/4.

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Step 4.

Extending the facts and investigations covered so far, we need to work the idea of a radius with respect to a square around circle.

The radius of the circle is 3. The diameter is 2R or 2 times the radius or 6. In this case the length of the line segment of our square is also 6, so the area would be 62 or 36.
Applying circle logic to this square, can you see that the area of this square in relation to the radius would be 4R2 or 4(9).

Simple thought- the area of the square is = 4R2 and the area of the circle is πR2, so the white area around the circle represents the difference between 4 and π.

What did we find? We can understand why Pi (π) is less than 4 and further consideration will help someone see why it is greater than 3. Take a look at this investigation again and then see if you can estimate the area of circle with any radius.
Sometimes a different point of view is all you need to make a connection to a concept.

Try some of these worksheets.