Welcome to Geometry for Beginners. This article returns to the concept of finding area, but this time the figure will be a circle rather than a polygon. Terms we have used previously for finding area - like base and height - do not apply to circles, so new terminology becomes necessary. In addition, we need to understand some concepts we have never encountered before to understand the derivation of the formula.
Note: Some mathematicians do, in fact, consider a circle to be a polygon - a polygon with an infinite number of sides. The concept of "infinite number of sides" comes from Calculus, but a few mental images can help Geometry students understand the basic idea. Get a piece of paper if your ability to visualize images in your mind is as weak as mine. Now, draw (either on paper or on your mental whiteboard) a triangle. With the triangle and all the other figures, try to make all the sides equal in length. Now, move to the right of the triangle and draw a square of similar size. Move right again, and draw a pentagon. Then draw a hexagon and an octagon. This is generally enough figures to see the pattern that as the number of sides increases, the polygon becomes more and more circular.
In Calculus, we consider what the "end result" would be if we could continue to increase the number of sides of a polygon forever. We call this end result the "limit." For our situation, a polygon with an infinite number of sides would have a circle as its limit.
In addition to understanding this limit concept, we also need to review the meaning of pi before we can understand the formula for area of a circle. Remember that the irrational number pi is the ratio of the circumference of a circle (distance around) to its diameter (distance across through the center). Also, remember that circumference is equivalent to the perimeter of polygons and has two possible formulas: C = (pi)d or C = 2(pi)r. Now we are ready to find the area of circles.
We already know that area is measured with squares; and, for rectangles, those squares are easy to see and count. Unfortunately, squares don't fit into circles nicely. To understand the area formula for circles, we need good mental image skills and a good understanding of the "limit" concept mentioned earlier in this article.
On your "paper" draw a circle with a diameter of 1 to 2 inches. Now, divide this circle into 4 equal parts by drawing another diameter perpendicular to the original diameter. You should now be able to see 4 shapes like pieces of pizza. Now, take those 4 pieces and fit them side by side but alternating point up and then point down. We now have a parallelogram-type figure having two bumps or curves on both the top and bottom and a rather steep lean to the side.
Now we are going to do the same type of limit process we discussed earlier. Look back at your circle with 4 parts. Draw two more diameters to divide each part in half. You should now see eight pie-shaped pieces that are the same "height" as before, but are more narrow. Take these eight pieces and fit them side by side, again alternating point up and point down. Again, we have that parallelogram-type shape, but now the lean to the side is decreased. Said in a different way, the sides are becoming more vertical. In addition, the top and bottom now have four bumps or curves each, but the curves are flatter.
As we continue to divide the circle into more and more pie pieces and continue fitting the pieces together side by side as before, the resulting figure becomes a rectangle because the sides become vertical and the curves on the top and bottom flatten completely. The height of this resulting rectangle is really the radius of the circle, r. The top and bottom of the rectangle come from the circumference. This means the base is one-half of the circumference, C.
The area of the circle is the same as the area of the rectangle. The rectangle area formula can, thus, change from A = bh to A = (1/2C)(r). Remembering the formula for circumference, we can change the area formula even further. A = (1/2C)(r) becomes A = 1/2(2(pi)r)(r). By simplifying the multiplication, the result is A = (pi)r^2.
This circle area formula, A = (pi)r^2, can be used to find the area if we know either the radius or diameter of the circle; or we can find what the radius or diameter must be for a given area.
For example: If the radius of a circle is 5 cm., find the area of the circle.
Solution: A = (pi)r^2 becomes A = (pi)5^2 or A = 25pi. The final form of the answer will depend on the teacher, the situation, or the subject. Sometimes, we want the answer in terms of pi because this is the EXACT answer, but we mentally estimate for meaning using 3 as the value of pi. Thus, the circle has an exact area of 25pi sq. cm. which is about 75 sq. cm. Other situations require a more accurate decimal value for the area, so we use the pi key on the calculator.
3 Final Cautions About Circles:
1. Answers with pi are EXACT, while decimals are APPROXIMATIONS.
2. Radius and diameter are often confused. Using the wrong value is very easy. THINK!
3. The circumference and area formulas are similar and easy to confuse. THINK before you start working on with a formula!
